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'''Positive semi-definite''' matrices are defined similarly, except that the scalars <math>\mathbf{x}^\top M \mathbf{x}</math> and <math>\mathbf{z}^* M \mathbf{z}</math> are required to be positive ''or zero'' (that is, nonnegative). '''Negative-definite''' and '''negative semi-definite''' matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''.
Some authors use more general definitions of definiteness, permitting the matrices to be non-symmetric or non-Hermitian. The properties of these generalized definite matrices are explored in the section [[#Extension for non-Hermitian square matrices|Extension for non-Hermitian square matrices]] below, but are not the main focus of this article.
== Ramifications ==
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