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

\mathbf{v}^\mathrm{H} A \mathbf{v} > 0

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.

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