Hermitian matrices (symmetric matrices)

Property: If $H$ is Hermitian, then:

1. $\forall x \in C^n, x^∗ H x \in R$;
2. Eigenvalues of H are real;
3. 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