Subsection5.4.1Hermitian Positive Definite matrices

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

Definition5.4.1.1. 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,$~^T \text{.}$

Example5.4.1.2.

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.

Homework5.4.1.1.

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

ALWAYS/SOMETIMES/NEVER: $A = B^H B$ is HPD.

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 $( B x )^H B x \gt 0 \text{.}$

Homework5.4.1.2.

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 $e_j \text{.}$

ALWAYS

Now prove it!

Solution

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

Homework5.4.1.3.

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.

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.