Positive-definite matrix

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).

Equivalent formulations

Let M be an n × n Hermitian matrix. In the following we denote the transpose of a matrix or 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:

\textbf{z}^{*} M \textbf{z} > 0

Note that the quantity

z^{*} M z

is always real.

\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.)

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

1.	For all non-zero vectors $z \in \mathbb{C}^n$ we have
2.	All eigenvalues $\lambda_i$ of $M$ are positive. (Recall that the eigenvalues of a Hermitian matrix are necessarily real.)
3.	The form
4.	All the following matrices (the leading principal minors) have a positive determinant (the Sylvester criterion):

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 the sum

M + N

and the products

MNM

and

NMN

are also positive definite; and if

M N = N M

, then

MN

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}

\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

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).

Matrices