ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5591 on D4937: Positive semidefinite real matrix
Subresult of R5590: Complex matrix gramians are positive semidefinite

Real matrix gramians are positive semidefinite

Formulation 0

Let $A \in \mathbb{R}^{N \times M}$ be a D4571: Real matrix.

Then

(1)	$A^T A$ is a D4937: Positive semidefinite real matrix
(2)	$A A^T$ is a D4937: Positive semidefinite real matrix

Proofs

Proof 0

Let $A \in \mathbb{R}^{N \times M}$ be a D4571: Real matrix.

This result is a particular case of R5590: Complex matrix gramians are positive semidefinite. $\square$