Symmetric positive matrix

Formal definition

$A \in S_{n}(\mathbb{R})$ is a symmetric positive matrix if $\forall x \in \mathbb{R}^{n}, x^{t}Ax \geq 0$.

$A$ is a definite symmetric positive matrix if $\forall x \in \mathbb{R}^{n}\setminus \{ 0\}, x^{t}Ax > 0$.

The set of symmetric positive matrices is written $S_{n}^{+}(\mathbb{R})$ while that of symmetric definite matrices is written $S_{n}^{++}(\mathbb{R})$.

Remarks

• Due to the Spectral Theorem, saying that a symmetric matrix $A$ is positive (resp. definite positive) is equivalent to saying that all its eigenvalues are positive (resp. strictly positive)