Prof. Matthieu Bloch
Monday, October 21, 2024 (v1.0)
Every complex matrix \(\matA\) has at least one complex eigenvector and every real symmetrix matrix has real eigenvalues and at least one real eigenvector.
Every matrix \(\matA\in\bbC^{n\times n}\) is unitarily similar to an upper triangular matrix, i.e., \[ \bfA = \bfV\boldsymbol{\Delta}\bfV^\dagger \] with \(\boldsymbol{\Delta}\) upper triangular and \(\bfV^\dagger=\bfV^{-1}\).
Every hermitian matrix is unitarily similar to a real-valued diagonal matrix.
A symmetric matrice \(\matA\) is positive definite if it has positive eigenvalues, i.e., \(\forall i\in\set{1,\cdots,n}\quad\lambda_i>0\).
A symmetric matrice \(\matA\) is positive semidefinite if it has nonnegative eigenvalues, i.e., \(\forall i\in\set{1,\cdots,n}\quad\lambda_i\geq 0\).
For any analytic function \(f\), we have \[ f(\matA) = \sum_{i=1}^n f(\lambda_i)\vecv_i\vecv_i^\intercal \]
Let \(\set{\vecv_i}\) be the eigenvectors of \(\matA\). \[ \vecx = \sum_{i=1}^n\frac{1}{\lambda_i}\dotp{\vecy}{\vecv_i}\vecv_i \]
\[ \frac{1}{\lambda_1^2}\norm{\vece}^2\leq \norm[2]{\vecx-\tilde{\vecx}}\leq \frac{1}{\lambda_n^2}\norm{\vece}^2. \]