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Subsection 5.4.1 Hermitian Positive Definite matrices

Hermitian Positive Definite (HPD) are a special class of matrices that are frequently encountered in practice.

Definition 5.4.1.1. Hermitian positive definite matrix.

A matrix \(A \in \C^{n \times n} \) is Hermitian positive definite (HPD) if and only if it is Hermitian (\(A^H = A\)) and for all nonzero vectors \(x \in \C^n \) it is the case that \(x ^H A x \gt 0 \text{.}\) If in addition \(A \in \R^{n \times n} \) then \(A \) is said to be symmetric positive definite (SPD).

If you feel uncomfortable with complex arithmetic, just replace the word "Hermitian" with "symmetric"" in this document and the Hermitian transpose operation,\(~^H \text{,}\) with the transpose operation,\(~^T \text{.}\)

Example 5.4.1.2.

Consider the case where \(n = 1 \) so that \(A \) is a real scalar, \(\alpha \text{.}\) Notice that then \(A \) is SPD if and only if \(\alpha \gt 0 \text{.}\) This is because then for all nonzero \(\chi \in \R \) it is the case that \(\alpha \chi^2 \gt 0 \text{.}\)

Let's get some practice with reasoning about Hermitian positive definite matrices.

Homework 5.4.1.1.

Let \(B \in \C^{m \times n} \) have linearly independent columns.

ALWAYS/SOMETIMES/NEVER: \(A = B^H B \) is HPD.

Answer

ALWAYS

Now prove it!

Solution

Let \(x \in \C^m \) be a nonzero vector. Then \(x^H B^H B x = ( B x )^H (B x ) \text{.}\) Since \(B \) has linearly independent columns we know that \(B x \neq 0 \text{.}\) Hence \(( B x )^H B x \gt 0 \text{.}\)

Homework 5.4.1.2.

Let \(A \in \C^{m \times m} \) be HPD.

ALWAYS/SOMETIMES/NEVER: The diagonal elements of \(A \) are real and positive.

Hint

Consider the standard basis vectors \(e_j \text{.}\)

Answer

ALWAYS

Now prove it!

Solution

Let \(e_j \) be the \(j \)th unit basis vectors. Then \(0 \lt e_j^H A e_j = \alpha_{j,j} \text{.}\)

Homework 5.4.1.3.

Let \(A \in \C^{m \times m} \) be HPD. Partition

\begin{equation*} A = \left( \begin{array}{c | c} \alpha_{11} \amp a_{21}^H \\ \hline a_{21} \amp A_{22} \end{array} \right). \end{equation*}

ALWAYS/SOMETIMES/NEVER: \(A_{22} \) is HPD.

Answer

ALWAYS

Now prove it!

Solution

We need to show that \(x_2^H A_{22} x_2 \gt 0 \) for any nonzero \(x_2 \in \C^{m-1}\text{.}\)

Let \(x_2 \in \C^{m-1} \) be a nonzero vector and choose \(x = \left( \begin{array}{c} 0 \\ x_2 \end{array} \right) \text{.}\) Then

\begin{equation*} \begin{array}{l} 0 \\ ~~~ \lt ~~~~ \lt A \mbox{ is HPD } \gt \\ x^H A x \\ ~~~ = ~~~~ \lt \mbox{ partition } \gt \\ \left( \begin{array}{c} 0 \\ x_2 \end{array} \right)^H \left( \begin{array}{c | c} \alpha_{11} \amp a_{21}^H \\ \hline a_{21} \amp A_{22} \end{array} \right) \left( \begin{array}{c} 0 \\ x_2 \end{array} \right) \\ ~~~ = ~~~ \lt \mbox{ multiply out } \gt \\ x_2^H A_{22} x_2 . \end{array} \end{equation*}

We conclude that \(A_{22} \) is HPD.