Hermitian matrix is a square matrix that equals its Hermitian adjoint: $A = A^H$. Symmetric matrix is a square matrix that equals its transpose: $A = A^T$. Real symmetric matrices are Hermitian; complex symmetric matrices are not. Similarly, skew-Hermitian matrix is a square matrix that negates its Hermitian adjoint: $A = - A^H$; skew-symmetric matrix is a square matrix that negates its transpose: $A = - A^T$.

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

Inequalities

Weyl inequalities of Hermitian eigenvalues (Thm 4.3.7): The eigenvalues of the sum of two Hermitian matrices are bounded above by the sums of the eigenvalues of the two of matching total order: $\lambda_i(S + T) \le \min_{j \le i} \lambda_j(S) + \lambda_{i-j+1}(T)$, $\forall S, T \in \mathcal{H}(n)$. Here, eigenvalues are in descending order, adopting the convention for singular values.

Corrolaries (Interlacing of Hermitian eigenvalues):

  • The change of each ordered eigenvalue cannot exceed the extreme eigenvalues of the perturbation: $\lambda_k(S) + \lambda_n(T) \le \lambda_k(S + T) \le \lambda_k(S) + \lambda_1(T)$;
  • Removing any paired row and column gives a spectrum that interlaces the original: $\lambda_{k+1}(S) \le \lambda_k(S_{-i}) \le \lambda_k(S), \forall i \in n, \forall k \in n-1$.
  • Positive rank-one perturbations increase each ordered eigenvalue, but not to exceed the next: $\lambda_k(S) \le \lambda_k(S + u u^T) \le \lambda_{k-1}(S), \forall u \in \mathbb{R}^n, \forall k \in n$.

🏷 Category=Algebra Category=Matrix Analysis