Eigenvalue Decomposition

20 Jan 2018 on Math

1. Eigenvalue and Eigenvector

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2. Example

- Matrix A

$A={\begin{bmatrix}2&1\\1&2\end{bmatrix} = 0}$

- Determinant of A

${\begin{aligned}|A-\lambda I|&=\left|{\begin{bmatrix}2&1\\1&2\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.\end{aligned}}$

- λ = 1 and λ = 3

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- when λ = 1

$(A-I)v_{\lambda =1}={\begin{bmatrix}1&1\\1&1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.$

- purple Vector = eigenvector

$v_{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}}$

- when λ = 3

$(A-3I)v_{\lambda =3}={\begin{bmatrix}-1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}$

- blue Vector = eigenvector

$v_{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}$

Eigenvalue of geometric transformation

Reference

Eigenvalues and eigenvectors - wikipedia