Lecture 3x Hermitian Matrices (pages 432-435) the complex equivalent of symmetric matrices. As usual, we add complex conjugation to the de nition A with complex entries is called He if A 2 1

Eigenvalue inequalities Denote by 1(A) n(A) the eigenvalues of a Hermitian matrix A 2 Mn. Theorem (Courant-Fischer-Weyl) Suppose A 2 Mn is Hermitian. Then for k 2 f1; : : : ; ng. (A) = ax min v Av = …

Observations: There are a number of quick computations that reveal interesting properties of Hermitian matrices. A is Hermitian if and only if iA is skew Hermitian. The diagonal entries of a Hermitian matrix …

Having navigated the complexity of nondiagonalizable matrices, we return for a closer examination of Hermitian matrices, a class whose mathematical elegance parallels its undeniable importance in a …

Hermitian Matrices h complex numbers. Let x = a + ib, where a; b are real umbers, and i = 1. Then, x = a ib is the comp ex conjugate of x. In the discussion below, all matrices and numbers are complex …

This allows us to extend the definitions of definite and semi-definite quadratic forms in real variables and real symmetric matrices to quadratic forms in complex variables and complex Hermitian matrices.

Spectral theorem for Hermitian matrices. For an Hermitian matrix, all eigenvalues are real, eigenvectors corresponding to distinct eigenvalues are orthogonal, there is an orthonormal basis consisting of …