Positive-definite matrix - Definition and Overview

In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. An n × n Hermitian matrix <math>M<math> is said to be positive definite if it has one (and therefore all) of the following 6 equivalent properties:

(1) For all non-zero vectors z in Cⁿ we have

<math>\textbf{z}^*M\textbf{z} > 0<math>.

Here we view z as a column vector with n complex entries and z^* as the complex conjugate of its transpose. (M being Hermitian, z^* M z is always real.)

(2) For all non-zero vectors x in Rⁿ we have

<math>\textbf{x}^TM\textbf{x} > 0<math>

(where x^T denotes the transpose of the column vector x).

(3) For all non-zero vectors u in Zⁿ (all components being integers), we have

<math>\textbf{u}^TM\textbf{u} > 0<math>.

(4) All eigenvalues of M are positive.

<math>\lambda_i(M) > 0 \; \forall_i<math>

(5) The form

<math>\langle\textbf{x},\textbf{y}\rangle = \textbf{x}^*M\textbf{y}<math>

defines an inner product on Cⁿ. (In fact, every inner product on Cⁿ arises in this fashion from a Hermitian positive definite matrix.)

(6) All the following matrices have positive determinant: 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, ..., and M itself.

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 rM is positive definite. If M and N are positive definite, then M + N is also positive definite, and if MN = NM, then MN is also positive definite. Every positive definite matrix M, has at least one square root matrix N such that N² = 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 Rⁿ (or, equivalently, all non-zero x in Cⁿ). It is called positive-semidefinite if

x^* M x ≥ 0

for all x in Rⁿ (or Cⁿ) and negative-semidefinite if

x^* M x ≤ 0

for all x in Rⁿ (or Cⁿ).

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

Generalizations

Suppose K denotes the field R or C, V is a vector space over K, and B : V × V → 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.