An n × n Hermitian matrix M is said to be positive definite if
- x* M x > 0
for all non-zero vectors x in Rn (or, equivalently, for all non-zero x in Cn). Here, we view x as a column vector and x* as the complex conjugate of its transpose.
Equivalently, a matrix M is positive definite if
- <x, y> = x* M y
defines an inner product on Rn (or, equivalently, on Cn).
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.