Eigenvalue and Eigenvector

Eigenvalues

Notation: $\lambda$

solve $\det (A-\lambda I)=0$
given by numpy.linalg.eighfor diagonal matrix and numpy.linalg.eig for non.

Singular value

Notation: $\delta$

${\delta }^2=\lambda$

Eigenvector

$$Ax=\lambda x$$

where $x$ is Eigenvector and $\lambda$ is Eigenvalues.

Applications

Problem transformations

• Shift, $(A-\sigma I)x=(\lambda -\sigma )x$ where $\sigma$ is scalar.
• Inversion, $A^{-1}x=\frac{x}{\lambda }$ if $A$ nonsingular and $x\ne 0$.
• Powers, $A^{k}x={\lambda }^{k}x$.
• Polynomial, $p(A)x=p(\lambda )x$ where $p(t)$ is polynomial.

Example

For $A$ is symmetric matrix, $e^A$ can be done by

$$e^A=X{e}^{\lambda }{X}^T$$

where $X$ is Eigenvector matrix

ev, v = np.linalg.eigh(A)
expmA = v @ np.diag(np.exp(ev)) @ v^T