In linear algebra, the positive-definite matrices are (in several ways) 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. (M being Hermitian, z* M z is always real - I think...)
(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 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.