Positive definite
Redirected from Positive semidefinite
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
(2) For all non-zero vectors x in
Rn we have
(3) For all non-zero vectors u in Zn (all components being integers), we have
(4) All eigenvalues of M are positive.
(5) The form
(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.
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.
The Hermitian matrix M is said to be negative definite if
for all non-zero x in Rn (or, equivalently, all non-zero x in Cn). It is called positive semidefinite if
for all x in Rn (or Cn) and negative semidefinite if
for all x in Rn (or Cn).
A Hermitian matrix which is neither positive nor negative semidefinite is called indefinite.
Here we view z as a column vector with n complex entries and z* as the complex conjugate of its transpose.
(where xT denotes the transpose of the column vector x).
defines an inner product on
Cn. (In fact, every inner product on Cn arises in this fashion from a Hermitian matrix.)
Further properties
Negative definite, semidefinite and indefinite matrices