A **positive-definite matrix** *A* is a Hermitian matrix that, for every non-zero column vector **v**,

where ^{H} is the conjugate transpose of **v**, which, in the case of only real numbers, is its transpose. A positive-definite matrix will have all positive eigenvalues. The identity matrix is an example of a positive definite matrix.

Negative definite, positive semi-definite, and negative semi-definite matrices are defined in a similar manner, with semi-definite matrices including zero.