Linear and multilinear algebra; matrix theory Definition

Positive definite matrix

Builds on Hermitian matrix Symmetric matrix

Definition

Let $A$ be an $n \times n$ complex Hermitian matrix. The matrix $A$ is positive definite when $x^* A x > 0 \qquad \text{for all } x \in \C^n \text{ with } x \ne 0.$ A real symmetric matrix $A$ is positive definite when $x^T A x > 0$ for all $x \in \R^n$ with $x \ne 0$ .

Remarks

For a Hermitian $A$ , the number $x^* A x$ is always real. The identity matrix and $\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$ are positive definite. $A$ is positive definite if and only if all its eigenvalues are positive. $A$ is positive definite if and only if $\langle x, y \rangle = y^* A x$ is an inner product on $\C^n$ .

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Matrix definition
Square matrix definition
Transpose definition
Symmetric matrix definition
Conjugate transpose definition
Hermitian matrix definition
Positive definite matrix definition

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12 pages build on this one.

Tag	`0QQCG3JWJ4`
URI	`m2://0QQCG3JWJ4`
Kind	definition
Writing	W1 stated
MSC	15B57, 15A63
Defines	Positive definite matrix
Uses	Hermitian matrix, Symmetric matrix

Positive definite matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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