Positivedefinite matrix

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In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form (or a sesquilinear form in the complex case).

1 Equivalent formulations
2 Further properties
3 Negative-definite, semidefinite and indefinite matrices
4 Non-Hermitian matrices
5 Generalizations
6 References

Equivalent formulations

Let M be an n × n Hermitian matrix. In the following we denote the transpose of a matrix of vector $a$ by $a T$ , and the conjugate transpose by $a *$ . The matrix M is said to be positive definite if it has one (and therefore all) of the following equivalent properties:

1.	For all non-zero vectors $z \in \mathbb{C}^n$ we have $\textbf{z}^{} M \textbf{z} > 0$ . Note that the quantity $z M z$ is always real.
2.	All eigenvalues $λ i$ of $M$ are positive. (Recall that the eigenvalues of a Hermitian matrix are necessarily real).
3.	The form $\langle \textbf{x},\textbf{y}\rangle = \textbf{x}^{*} M \textbf{y}$ defines an inner product on $\mathbb{C}^n$ . (In fact, every inner product on $\mathbb{C}^n$ arises in this fashion from a Hermitian positive definite matrix.)
4.	All the following matrices (the leading principle minors) have a positive determinant (the Sylvester criterion): the upper left 1-by-1 corner of $M$ the upper left 2-by-2 corner of $M$ the upper left 3-by-3 corner of $M$ ... $M$ itself

Analogous statements hold if M is a real symmetric matrix, by replacing $\mathbb{C}^n$ by $\mathbb{R}^n$ , and the conjugate transpose by the transpose.

Further properties

Every positive definite matrix is invertible and its inverse is also positive definite. If $M$ is positive definite and $r > 0$ is a real number, then $r M$ is positive definite. If $M$ and $N$ are positive definite, then $M + N$ is also positive definite, and if $M N = N M$ , then $M N$ is also positive definite. Every positive definite matrix $M$ , has at least one square root matrix $N$ such that $N 2 = M$ . In fact, $M$ may have infinitely many square roots, but exactly one positive definite square root.

Negative-definite, semidefinite and indefinite matrices

The Hermitian matrix $M$ is said to be negative-definite if

$x^{*} M x < 0\,$

for all non-zero $x \in \mathbb{R}^n$ (or, equivalently, all non-zero $x \in \mathbb{C}^n$ ). It is called positive-semidefinite if

$x^{*} M x \geq 0$

for all $x \in \mathbb{R}^n$ (or $\mathbb{C}^n$ ) and negative-semidefinite if

$x^{*} M x \leq 0$

for all $x \in \mathbb{R}^n$ (or $\mathbb{C}^n$ ).

A Hermitian matrix which is neither positive- nor negative-semidefinite is called indefinite.

Non-Hermitian matrices

A real matrix M may have the property that x^TMx > 0 for all nonzero real vectors x without being symmetric. The matrix

$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$

provides an example. In general, we have x^TMx > 0 for all real nonzero vectors x if and only if the symmetric part, (M + M^T) / 2, is positive definite.

The situation for complex matrices may be different, depending on how one generalizes the inequality z^*Mz > 0. If z^*Mz is real for all complex vectors z, then the matrix M is necessarily Hermitian. So, if we require that z^*Mz be real and positive, then M is automatically Hermitian. On the other hand, we have that Re(z^*Mz) > 0 for all complex nonzero vectors z if and only if the Hermitian part, (M + M_*) / 2, is positive definite.

There is no agreement in the literature on the proper definition of positive-definite for non-Hermitian matrices.

Generalizations

Suppose $K$ denotes the field $\mathbb{R}$ or $\mathbb{C}$ , $V$ is a vector space over $K$ , and $B : V \times V \rightarrow K$ is a bilinear map which is Hermitian in the sense that $B (x, y)$ is always the complex conjugate of $B (y, x)$ . Then $B$ is called positive definite if $B (x, x) > 0$ for every nonzero $x$ in $V$ .

References

Roger A. Horn and Charles R. Johnson. Matrix Analysis, Chapter 7. Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).fr:Matrice définie positive

it:Matrice definita positiva he:מטריצה חיובית

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Category: Matrices