Definition 28 \(\tilde A\;N\times N\) matrix \(\vec{x}\;N\times 1\) vector is called Eigenvector if
| \[\tilde A\vec{x}=\lambda\vec{x}\] |
where the \(\lambda\in\mathbb{R}\;\mbox{or }\mathbb{C}\) are called Eigenvalues of \(\tilde A\).
Example \begin{eqnarray*}\tilde A&=&\left(\begin{array}{cc}1&-2\\ -2&1\end{array}\right)\qquad\vec{x}=\vect{1\\1}\\\\ \tilde A\vec{x}&=&\vect{-1\\-1}=-1\cdot\vec{x}\;\Rightarrow\;\begin{array}{ll}\vec{x}&\mbox{Eigenvector}\\\lambda=-1&\mbox{Eigenvalue}\end{array}\end{eqnarray*}
General treatment \begin{eqnarray*}\tilde A\vec{x}=\lambda\vec{x}&\Leftrightarrow&\underbrace{\left(\tilde A-\lambda \tilde I\right)}_{\mbox{homogeneous system for $\vec{x}$}}\vec{x}=\vec{0}\\ &\Rightarrow&\mbox{only soluble if } \det(\tilde A-\lambda\tilde I)=0\end{eqnarray*}
Thus:
| \[ \begin{array}{cll} \left|\begin{array}{cccl} a_{11}-\lambda&a_{12}&\cdots&a_{1N}\\ a_{21}&a_{22}-\lambda& &a_{2N}\\ \vdots& &\ddots&\vdots\\ a_{N1}&a_{N2}&\cdots&a_{NN}-\lambda \end{array}\right|&=0&\quad\quad\begin{array}{ll}\mbox{ {Def.:}}&\mbox{$P(\lambda)$ is the characteristic polynomial}\\ &\mbox{associated with $\tilde A$}\end{array} \end{array}\] |
| \[P(\lambda)=(-1)^N\lambda^N+\alpha_{N-1}\lambda^{N-1}+\ldots+\alpha_1\lambda+\alpha_0=0\] |
characteristic polynomial!!
Polynomial equation for \(\lambda\)
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© J. Carstensen (Math for MS)