Ivan Shishkin, Birch Grove

Symmetric positive matrix

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Revision 3391

9/1/2026, 10:20:47 AM · Catalpa

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1##### Formal definition
1$A \in S_{n}(\mathbb{R})$ is a symmetric positive matrix if $\forall x \in \mathbb{R}^{n}, x^{t}Ax \geq 0$.
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3$A$ is a definite symmetric positive matrix if $\forall x \in \mathbb{R}^{n}\setminus \{ 0\}, x^{t}Ax > 0$.
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5The set of symmetric positive matrices is written $S_{n}^{+}(\mathbb{R})$ while that of symmetric definite matrices is written $S_{n}^{++}(\mathbb{R})$.
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7Due to the [[Spectral Theorem|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)
8##### Remarks
9- Due to the [[Spectral Theorem|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)

Revision 2299

8/25/2026, 11:50:47 AM · Anduril

Updated linked exercises

linked exercisesNoneEigenvalues of positive matrices

Revision 2298

8/25/2026, 11:50:15 AM · Anduril

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Compare with revision 22951 changed line
1$A \in S_{n}(\mathbb{R})$ is a symmetric positive matrix if $\forall x \in \mathbb{R}^{n}, x^{t}Ax \geq 0$.
2
3$A$ is a definite symmetric positive matrix if $\forall x \in \mathbb{R}^{n}\setminus \{ 0\}, x^{t}Ax > 0$.
4
5The set of symmetric positive matrices is written $S_{n}^{+}(\mathbb{R})$ while that of symmetric definite matrices is written $S_{n}^{++}(\mathbb{R})$.
5
6Due to the [[Spectral Theorem|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)

Revision 2295

8/25/2026, 11:42:01 AM · Anduril

Concept created

$A \in S_{n}(\mathbb{R})$ 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 $\forall x \in \mathbb{R}^{n}\setminus \{ 0\}, x^{t}Ax > 0$.


Due to the [[Spectral Theorem|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)