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Positive definite

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The positive definite matrices are in several senses analogous to the positive real numbers. An n × n Hermitian matrix M 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

z* M z > 0.
Here we view z as a column vector with n complex entries and z* as the complex conjugate of its transpose.

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

xT M x > 0
(where xT denotes the transpose of the column vector x).

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

uT M u > 0.

(4) All eigenvalues of M are positive.

(5) The form

<x, y> = x* M y
defines an inner product on Cn. (In fact, every inner product on Cn arises in this fashion from a Hermitian 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. To every positive definite matrix M, there exists precisely one square root: a positive definite matrix N with N2 = M.

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.

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