Internal problem ID [6718]
Internal file name [OUTPUT/5966_Sunday_June_05_2022_04_07_58_PM_25711530/index.tex]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.
EXERCISES 8.1. Page 332
Problem number: 8.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} x^{\prime }\left (t \right )&=7 x \left (t \right )+5 y-9 z \left (t \right )-8 \,{\mathrm e}^{-2 t}\\ y^{\prime }&=4 x \left (t \right )+y+z \left (t \right )+2 \,{\mathrm e}^{5 t}\\ z^{\prime }\left (t \right )&=-2 y+3 z \left (t \right )+{\mathrm e}^{5 t}-3 \,{\mathrm e}^{-2 t} \end {align*}
In this method, we will assume we have found the matrix exponential \(e^{A t}\) allready. There are different methods to determine this but will not be shown here. This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) + \vec {G}(t) \end {align*}
Or \begin {align*} \left [\begin {array}{c} x^{\prime }\left (t \right ) \\ y^{\prime } \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} 7 & 5 & -9 \\ 4 & 1 & 1 \\ 0 & -2 & 3 \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] + \left [\begin {array}{c} -8 \,{\mathrm e}^{-2 t} \\ 2 \,{\mathrm e}^{5 t} \\ {\mathrm e}^{5 t}-3 \,{\mathrm e}^{-2 t} \end {array}\right ] \end {align*}
Since the system is nonhomogeneous, then the solution is given by \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p(t) \end {align*}
Where \(\vec {x}_h(t)\) is the homogeneous solution to \(\vec {x}'(t) = A\, \vec {x}(t)\) and \(\vec {x}_p(t)\) is a particular solution to \(\vec {x}'(t) = A\, \vec {x}(t) + \vec {G}(t)\). The particular solution will be found using variation of parameters method applied to the fundamental matrix. For the above matrix \(A\), the matrix exponential can be found to be \begin {align*} e^{A t} &= \text {Expression too large to display}\\ &= \text {Expression too large to display} \end {align*}
Therefore the homogeneous solution is \begin {align*} \vec {x}_h(t) &= e^{A t} \vec {c} \\ &= \text {Expression too large to display} \left [\begin {array}{c} c_{1} \\ c_{2} \\ c_{3} \end {array}\right ] \\ &= \text {Expression too large to display}\\ &= \text {Expression too large to display} \end {align*}
The particular solution given by \begin {align*} \vec {x}_p (t) &= e^{A t} \int { e^{-A t} \vec {G}(t) \,dt} \end {align*}
But \begin {align*} e^{-A t} &= (e^{A t})^{-1} \\ &= \text {Expression too large to display} \end {align*}
Hence \begin {align*} \vec {x}_p (t) &= \text {Expression too large to display} \int { \text {Expression too large to display} \left [\begin {array}{c} -8 \,{\mathrm e}^{-2 t} \\ 2 \,{\mathrm e}^{5 t} \\ {\mathrm e}^{5 t}-3 \,{\mathrm e}^{-2 t} \end {array}\right ]\,dt}\\ &= \text {Expression too large to display} \text {Expression too large to display}\\ &= \left [\begin {array}{c} -\frac {\left (3125 \,{\mathrm e}^{7 t}-30624\right ) {\mathrm e}^{-2 t}}{16500} \\ \frac {\left (575 \,{\mathrm e}^{7 t}-3894\right ) {\mathrm e}^{-2 t}}{1650} \\ \frac {\left (625 \,{\mathrm e}^{7 t}-1419\right ) {\mathrm e}^{-2 t}}{4125} \end {array}\right ] \end {align*}
Hence the complete solution is \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p (t) \\ &= \text {Expression too large to display} \end {align*}
This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) + \vec {G}(t) \end {align*}
Or \begin {align*} \left [\begin {array}{c} x^{\prime }\left (t \right ) \\ y^{\prime } \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} 7 & 5 & -9 \\ 4 & 1 & 1 \\ 0 & -2 & 3 \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] + \left [\begin {array}{c} -8 \,{\mathrm e}^{-2 t} \\ 2 \,{\mathrm e}^{5 t} \\ {\mathrm e}^{5 t}-3 \,{\mathrm e}^{-2 t} \end {array}\right ] \end {align*}
Since the system is nonhomogeneous, then the solution is given by \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p(t) \end {align*}
Where \(\vec {x}_h(t)\) is the homogeneous solution to \(\vec {x}'(t) = A\, \vec {x}(t)\) and \(\vec {x}_p(t)\) is a particular solution to \(\vec {x}'(t) = A\, \vec {x}(t) + \vec {G}(t)\). The particular solution will be found using variation of parameters method applied to the fundamental matrix.
The first step is find the homogeneous solution. We start by finding the eigenvalues of \(A\). This is done by solving the following equation for the eigenvalues \(\lambda \) \begin {align*} \operatorname {det} \left ( A- \lambda I \right ) &= 0 \end {align*}
Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} 7 & 5 & -9 \\ 4 & 1 & 1 \\ 0 & -2 & 3 \end {array}\right ]-\lambda \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}
Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} 7-\lambda & 5 & -9 \\ 4 & 1-\lambda & 1 \\ 0 & -2 & 3-\lambda \end {array}\right ]\right ) &= 0 \end {align*}
Which gives the characteristic equation \begin {align*} \lambda ^{3}-11 \lambda ^{2}+13 \lambda -47&=0 \end {align*}
The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= \frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\\ \lambda _2 &= -\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
This table summarises the above result
| eigenvalue | algebraic multiplicity | type of eigenvalue |
| \(-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
| \(\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\) | \(1\) | real eigenvalue |
| \(-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
Now the eigenvector for each eigenvalue are found.
Considering the eigenvalue \(\lambda _{1} = \frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 7 & 5 & -9 \\ 4 & 1 & 1 \\ 0 & -2 & 3 \end {array}\right ] - \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}} & 5 & -9 \\ 4 & \frac {-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-8 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}} & 1 \\ 0 & -2 & \frac {-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-2 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {10}{3}-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}&5&-9&0\\ 4&-\frac {8}{3}-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}&1&0\\ 0&-2&-\frac {2}{3}-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {4 R_{1}}{\frac {10}{3}-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}&5&-9&0\\ 0&\frac {2 \left (\sqrt {33231}+193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-4 \sqrt {33231}-32 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1360}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82\right )}&\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-118 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82}{\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82}&0\\ 0&-2&\frac {-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-2 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82\right ) R_{2}}{\left (\sqrt {33231}+193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2 \sqrt {33231}-16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+680} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}&5&-9&0\\ 0&\frac {2 \left (\sqrt {33231}+193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-4 \sqrt {33231}-32 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1360}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82\right )}&\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-118 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82}{\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}} & 5 & -9 \\ 0 & \frac {2 \left (\sqrt {33231}+193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-4 \sqrt {33231}-32 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1360}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82\right )} & \frac {\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-118 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82}{\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )}, v_{2} = \frac {t \left (59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661\right )}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680}\right \}\)
Hence the solution is \[ \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {t \left (59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661\right )}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {t \left (59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661\right )}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {t \left (59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661\right )}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {t \left (59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661\right )}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {t \left (59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661\right )}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (9 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+151 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-33 \sqrt {33231}+1532 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2815\right )}{\left (16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680\right ) \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-10 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ \frac {59 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-41 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-3 \sqrt {33231}-661}{16 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-193 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \sqrt {33231}-680} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{2} = -\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 7 & 5 & -9 \\ 4 & 1 & 1 \\ 0 & -2 & 3 \end {array}\right ] - \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} & 5 & -9 \\ 4 & \frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}-16 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} & 1 \\ 0 & -2 & \frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}-4 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {10}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}&5&-9&0\\ 4&-\frac {8}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}&1&0\\ 0&-2&-\frac {2}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {4 R_{1}}{\frac {10}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&5&-9&0\\ 0&\frac {2 \left (193 i \sqrt {3}+3 i \sqrt {11077}-\sqrt {33231}-193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-1360 i \sqrt {3}+12 i \sqrt {11077}+4 \sqrt {33231}-64 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-1360}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (82+20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+i \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82\right ) \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}\right )}&\frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+236 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}&0\\ 0&-2&\frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}-4 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (82+20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+i \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82\right ) \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}\right ) R_{2}}{\left (193 i \sqrt {3}+3 i \sqrt {11077}-\sqrt {33231}-193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-680 i \sqrt {3}+6 i \sqrt {11077}+2 \sqrt {33231}-32 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-680} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&5&-9&0\\ 0&\frac {2 \left (193 i \sqrt {3}+3 i \sqrt {11077}-\sqrt {33231}-193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-1360 i \sqrt {3}+12 i \sqrt {11077}+4 \sqrt {33231}-64 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-1360}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (82+20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+i \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82\right ) \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}\right )}&\frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+236 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} & 5 & -9 \\ 0 & \frac {2 \left (193 i \sqrt {3}+3 i \sqrt {11077}-\sqrt {33231}-193\right ) \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-1360 i \sqrt {3}+12 i \sqrt {11077}+4 \sqrt {33231}-64 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-1360}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (82+20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+i \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82\right ) \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}\right )} & \frac {i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+236 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (-1974 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-41 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-123 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-658 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-99032 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+6216 \sqrt {33231}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-99032 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+1369592\right )}{\left (-3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+1092 \sqrt {33231}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right ) \left (i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )}, v_{2} = -\frac {t \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+1529 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-3 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-128 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-27082 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-384 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2688 \sqrt {33231}-1529 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-27082 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-592256\right )}{2 \left (-3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+1092 \sqrt {33231}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right )}\right \}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (-1974 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-41 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-123 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-658 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-99032 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+6216 \sqrt {33231}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-99032 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+1369592\right )}{\left (-3 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-427 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-6 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2021 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+1092 \sqrt {33231}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right ) \left (\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 \,\operatorname {I} \sqrt {3}+20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ -\frac {t \left (\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+1529 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-3 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-128 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-27082 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-384 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2688 \sqrt {33231}-1529 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-27082 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-592256\right )}{2 \left (-3 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-427 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-6 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2021 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+1092 \sqrt {33231}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right )} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (-1974 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-41 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-123 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-658 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-99032 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+6216 \sqrt {33231}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-99032 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+1369592\right )}{\left (-3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+1092 \sqrt {33231}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right ) \left (i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 i \sqrt {3}+20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+82\right )} \\ -\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+1529 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-3 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-128 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-27082 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-384 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2688 \sqrt {33231}-1529 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-27082 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-592256}{2 \left (-3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}-427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+1092 \sqrt {33231}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right )} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{3} = -\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 7 & 5 & -9 \\ 4 & 1 & 1 \\ 0 & -2 & 3 \end {array}\right ] - \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {-i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} & 5 & -9 \\ 4 & \frac {-i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+82 i \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-16 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} & 1 \\ 0 & -2 & \frac {-i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+82 i \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-4 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {10}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}&5&-9&0\\ 4&-\frac {8}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}&1&0\\ 0&-2&-\frac {2}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {4 R_{1}}{\frac {10}{3}+\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}+\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&5&-9&0\\ 0&\frac {386 \left (\left (i+\frac {\sqrt {11077}}{193}\right ) \sqrt {3}+\frac {3 i \sqrt {11077}}{193}+1\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-1360 i \sqrt {3}+12 i \sqrt {11077}-4 \sqrt {3}\, \sqrt {11077}+64 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+1360}{\left (i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-82\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&\frac {i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-82 i \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-236 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}&0\\ 0&-2&\frac {-i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+82 i \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-4 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}+\frac {\left (i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-82\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}} R_{2}}{193 \left (\left (i+\frac {\sqrt {11077}}{193}\right ) \sqrt {3}+\frac {3 i \sqrt {11077}}{193}+1\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-680 i \sqrt {3}+6 i \sqrt {11077}-2 \sqrt {3}\, \sqrt {11077}+32 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+680} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&5&-9&0\\ 0&\frac {386 \left (\left (i+\frac {\sqrt {11077}}{193}\right ) \sqrt {3}+\frac {3 i \sqrt {11077}}{193}+1\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-1360 i \sqrt {3}+12 i \sqrt {11077}-4 \sqrt {3}\, \sqrt {11077}+64 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+1360}{\left (i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-82\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}}&\frac {i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-82 i \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-236 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+82 i \sqrt {3}+20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}+82}{6 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} & 5 & -9 \\ 0 & \frac {386 \left (\left (i+\frac {\sqrt {11077}}{193}\right ) \sqrt {3}+\frac {3 i \sqrt {11077}}{193}+1\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-1360 i \sqrt {3}+12 i \sqrt {11077}-4 \sqrt {3}\, \sqrt {11077}+64 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}+1360}{\left (i \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}-82\right ) \left (1322+6 \sqrt {3}\, \sqrt {11077}\right )^{\frac {1}{3}}} & \frac {i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-82 i \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-236 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82}{i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = -\frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (123 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}+41 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+1974 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+99032 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-658 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+6216 \sqrt {33231}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-99032 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+1369592\right )}{\left (2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1092 \sqrt {33231}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right ) \left (i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82\right )}, v_{2} = -\frac {t \left (384 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+3 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-1529 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+27082 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-128 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2688 \sqrt {33231}-1529 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-27082 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-592256\right )}{2 \left (2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1092 \sqrt {33231}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right )}\right \}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} -\frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} t \left (123 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}+41 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+1974 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+99032 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-658 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+6216 \sqrt {33231}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-99032 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+1369592\right )}{\left (2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+427 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+6 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2021 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+3 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1092 \sqrt {33231}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right ) \left (\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 \,\operatorname {I} \sqrt {3}-20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82\right )} \\ -\frac {t \left (384 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+3 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-1529 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+27082 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-128 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2688 \sqrt {33231}-1529 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-27082 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-592256\right )}{2 \left (2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+427 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+6 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2021 \,\operatorname {I} \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+3 \,\operatorname {I} \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1092 \sqrt {33231}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right )} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \left (123 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}+41 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+1974 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+99032 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-658 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+6216 \sqrt {33231}+41 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-99032 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+1369592\right )}{\left (2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1092 \sqrt {33231}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right ) \left (i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}-\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-82 i \sqrt {3}-20 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-82\right )} \\ -\frac {384 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+3 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-1529 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+\left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+27082 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}-128 \sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2688 \sqrt {33231}-1529 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}-27082 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-592256}{2 \left (2 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {33231}+427 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {3}+6 i \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}} \sqrt {11077}-\sqrt {33231}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-2021 i \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {11077}\, \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+427 \left (1322+6 \sqrt {33231}\right )^{\frac {2}{3}}+1092 \sqrt {33231}+2021 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+240604\right )} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity \(m\), and its geometric multiplicity \(k\) and the eigenvectors associated with the eigenvalue. If \(m>k\) then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity \(k\)) does not equal the algebraic multiplicity \(m\), and we need to determine an additional \(m-k\) generalized eigenvectors for this eigenvalue.
| multiplicity
| ||||
| eigenvalue | algebraic \(m\) | geometric \(k\) | defective? | eigenvectors |
| \(\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {114 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+\frac {9348}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+722}{\left (5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}\right ) \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {10}{3}\right )} \\ \frac {9 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-12 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-\frac {984}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-305}{5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-57 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-\frac {4674}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+722+171 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{\left (5 \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {29 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {2378}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}-29 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )\right ) \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {10}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+6 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+\frac {492}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-305-18 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{5 \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {29 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {2378}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}-29 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-57 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-\frac {4674}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+722-171 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{\left (5 \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {29 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {2378}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}+29 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )\right ) \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {10}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+6 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+\frac {492}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-305+18 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{5 \left (-\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{6}-\frac {41}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {29 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {2378}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}+29 i \sqrt {3}\, \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}\right )} \\ 1 \end {array}\right ]\) |
Now that we found the eigenvalues and associated eigenvectors, we will go over each eigenvalue and generate the solution basis. The only problem we need to take care of is if the eigenvalue is defective. Since eigenvalue \(\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t}\\ &= \left [\begin {array}{c} \frac {114 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+\frac {9348}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+722}{\left (5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}\right ) \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {10}{3}\right )} \\ \frac {9 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-12 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-\frac {984}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-305}{5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}} \\ 1 \end {array}\right ] e^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t} \end {align*}
Therefore the homogeneous solution is \begin {align*} \vec {x}_h(t) &= c_{1} \vec {x}_{1}(t) + c_{2} \vec {x}_{2}(t) + c_{3} \vec {x}_{3}(t) \end {align*}
Which is written as \begin {align*} \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} \frac {38 \,{\mathrm e}^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t} \left (3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+\frac {246}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+19\right )}{\left (5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}\right ) \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {10}{3}\right )} \\ \frac {{\mathrm e}^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t} \left (9 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-12 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-\frac {984}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-305\right )}{5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}} \\ {\mathrm e}^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t} \end {array}\right ] + c_{2} \text {Expression too large to display} + c_{3} \text {Expression too large to display} \end {align*}
Now that we found homogeneous solution above, we need to find a particular solution \(\vec {x}_p(t)\). We will use Variation of parameters. The fundamental matrix is \[ \Phi =\begin {bmatrix} \vec {x}_{1} & \vec {x}_{2} & \cdots \end {bmatrix} \] Where \(\vec {x}_i\) are the solution basis found above. Therefore the fundamental matrix is \begin {align*} \Phi (t)&= \text {Expression too large to display} \end {align*}
The particular solution is then given by \begin {align*} \vec {x}_p(t) &= \Phi \int { \Phi ^{-1} \vec {G}(t) \, dt}\\ \end {align*}
But \begin {align*} \Phi ^{-1} &= \text {Expression too large to display} \end {align*}
Hence \begin {align*} \vec {x}_p(t) &= \text {Expression too large to display} \int { \text {Expression too large to display} \left [\begin {array}{c} -8 \,{\mathrm e}^{-2 t} \\ 2 \,{\mathrm e}^{5 t} \\ {\mathrm e}^{5 t}-3 \,{\mathrm e}^{-2 t} \end {array}\right ] \, dt}\\ &= \text {Expression too large to display} \int { \text {Expression too large to display} \, dt}\\ &= \text {Expression too large to display} \text {Expression too large to display} \\ &= \left [\begin {array}{c} -\frac {\left (3125 \,{\mathrm e}^{7 t}-30624\right ) {\mathrm e}^{-2 t}}{16500} \\ \frac {\left (575 \,{\mathrm e}^{7 t}-3894\right ) {\mathrm e}^{-2 t}}{1650} \\ \frac {\left (625 \,{\mathrm e}^{7 t}-1419\right ) {\mathrm e}^{-2 t}}{4125} \end {array}\right ] \end {align*}
Now that we found particular solution, the final solution is \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p(t)\\ \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] &= \left [\begin {array}{c} \frac {38 c_{1} {\mathrm e}^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t} \left (3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}+\frac {246}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+19\right )}{\left (5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}\right ) \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {10}{3}\right )} \\ \frac {c_{1} {\mathrm e}^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t} \left (9 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-12 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}-\frac {984}{\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-305\right )}{5 \left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right )^{2}-\frac {58 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}-\frac {4756}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}-\frac {749}{3}} \\ c_{1} {\mathrm e}^{\left (\frac {\left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}{3}+\frac {82}{3 \left (1322+6 \sqrt {33231}\right )^{\frac {1}{3}}}+\frac {11}{3}\right ) t} \end {array}\right ] + \text {Expression too large to display} + \text {Expression too large to display} + \left [\begin {array}{c} -\frac {\left (3125 \,{\mathrm e}^{7 t}-30624\right ) {\mathrm e}^{-2 t}}{16500} \\ \frac {\left (575 \,{\mathrm e}^{7 t}-3894\right ) {\mathrm e}^{-2 t}}{1650} \\ \frac {\left (625 \,{\mathrm e}^{7 t}-1419\right ) {\mathrm e}^{-2 t}}{4125} \end {array}\right ] \end {align*}
Which becomes \begin {align*} \text {Expression too large to display} \end {align*}
✓ Solution by Maple
Time used: 5.688 (sec). Leaf size: 9313
dsolve([diff(x(t),t)=7*x(t)+5*y(t)-9*z(t)-8*exp(-2*t),diff(y(t),t)=4*x(t)+y(t)+z(t)+2*exp(5*t),diff(z(t),t)=-2*y(t)+3*z(t)+exp(5*t)-3*exp(-2*t)],singsol=all)
\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.206 (sec). Leaf size: 3002
DSolve[{x'[t]==7*x[t]+5*y[t]-9*z[t]-8*Exp[-2*t],y'[t]==4*x[t]+y[t]+z[t]+2*Exp[5*t],z'[t]==-2*y[t]+3*z[t]+Exp[5*t]-3*Exp[-2*t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
Too large to display