Hermitian matrices (symmetric matrices)

Property:
If $H$ is Hermitian, then:

- $\forall x \in C^n, x^∗ H x \in R$;
- Eigenvalues of H are real;
- Eigenvectors associated with different eigenvalues are orthogonal.
(By orthogonal in C, we mean $x^∗ y = 0$)

Theorem (**spectral theorem**, **principal axis theorem**):
A Hermitian (symmetric) can be diagonalized by unitary (orthogonal) matrix.

(Matrices equivalent to a Hermitian matrix is Hermitian. ?)

Note: A real matrix with real eigenvalues has real eigenvectors,
they are not orthogonal unless A is symmetric.

Property:
If A Hermitian and $x^∗ A x \succeq 0 \forall x \in C_n$,
then all eigenvalues of A are (real) nonnegative.

Property:
Every $A \in M_n(C)$ has a unique decomposition as $A = H + iH'$, with H and H' Hermitian.

Theorem (proof not shown):
Let F be a family of Hermitian matrices,
then exist U unitary s.t. $U^∗ A U$ diagonal for any A in F iff F is a commuting family.

**Rayleigh-Ritz ratio**: ${x^∗ A x} / {x^∗ x}$

Theorem (**Rayleigh-Ritz**)

## Partial Ordering of Hermitian Matrices

Notes on Partial Ordering of Hermitian Matrices

## Inequalities

🏷 Category=Algebra Category=Matrix Theory