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Real Positive Semidefinite Matrix Cone
Needed by:
Monotonic Functions of Real Matrices
Multivariate Normal Entropy
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Real Positive Semidefinite Matrix Order


Can we order the cone of positive semidefinite matrices?


The positive semidefinite matrix order (or Loewner order) is a partial ordering $\geq$ on $\mathbfsf{S} ^d$ defined by

\[ A \geq B \quad \iff \quad A - B \geq 0 \quad \iff \quad A - B \in \mathbfsf{S} _+^d. \]

We define the partial order $>$ on symmetric matrices by

\[ A > B \quad \iff \quad A - B > 0 \quad \iff \quad A - B \in \mathbfsf{S} _{++}^d. \]


Each of the following results from the geometric properties of the positive semidefinite cone:

\[ \begin{aligned} \alpha A \geq 0 \quad & \text{ for all } \delta > 0, A \geq 0, \\ A + B \geq 0 \quad & \text{ for all } A, B \geq 0, \\ A \geq B \text{ and } B \geq A \implies A = B \quad & \text{ for all } A, B \in \mathbfsf{S} ^d, \\ \lim_{n \to \infty} A_n = A \implies A \geq 0 \quad & \text{ for all sequences } (A_n)_n \text{ in } \mathbfsf{S} _+^d. \end{aligned} \]

Partial Order

$A \geq B$ and $B \geq A$ giving $A = B$ means that $\geq$ is antisymmetric. Moreover,

\[ \begin{aligned} A \geq A \quad & \text{ for all } A \in \mathbfsf{S} ^d, \text{ and } \\ A \geq B \text{ and } B \geq C \implies A \geq C & \text { for all } A, B, C \in \mathbfsf{S} ^d. \end{aligned} \]

In other words, $\geq$ is also reflexive and transitive. In other words, $\geq$ is a partial order (see  Orders).1

For $d = 1$, $\geq$ reduces to the familiar total order of the real line (see Real Order). The converse perspective is to see the positive semidefinite order as an extension of the order on $\R $ to the space $\mathbfsf{S} ^d$. Of course, the key difference is that two matrices may not be comparable. The order is partial.

For example, the matrices $A, B \in \mathbfsf{S} ^2$ defined by

\[ A = \bmat{ 1 & 0 \\ 0 & 0 }, \quad B = \bmat{ 0 & 0 \\ 0 & 1 } \]

are not comparable. Neither $A \geq B$ nor $B \geq A$ holds.

  1. Future editions will include more formal accounts. ↩︎
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