Chapter 9

Hermitian and Positive Definite Matrices

Wayne Barrett

Brigham Young University

Information on self-adjoint (i.e., Hermitian) and positive definite linear operators (usually called positive linear operators in the literature) can be found in Section 5.3. For a finite dimensional inner product space, an orthonormal basis provides a correspondence between matrix results and linear operators (see Fact 5.3.6).

9.1 Hermitian Matrices

All matrices in this section are either real or complex, unless explicitly stated otherwise.

Definitions:

A matrix A ∈ ℂn×n is Hermitian or self-adjoint if A* = A, or element-wise, āij = aji, for i, j = 1, ..., n. The set of Hermitian matrices of order n is denoted by Hn. Note that a matrix A∈ ℝn×n is Hermitian ...

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