Definite matrix

In mathematics, a symmetric matrix ${\displaystyle M}$ with real entries is positive-definite if the real number ${\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} }$ is positive for every nonzero real column vector ${\displaystyle \mathbf {x} ,}$ where ${\displaystyle \mathbf {x} ^{\mathsf {T}}}$ is the row vector transpose of ${\displaystyle \mathbf {x} .}$ More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number ${\displaystyle \mathbf {z} ^{*}M\mathbf {z} }$ is positive for every nonzero complex column vector ${\displaystyle \mathbf {z} ,}$ where ${\displaystyle \mathbf {z} ^{*}}$ denotes the conjugate transpose of ${\displaystyle \mathbf {z} .}$

Positive semi-definite matrices are defined similarly, except that the scalars ${\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} }$ and ${\displaystyle \mathbf {z} ^{*}M\mathbf {z} }$ 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 § Extension for non-Hermitian square matrices, below, but are not the main focus of this article.

In the following definitions, ${\displaystyle \mathbf {x} ^{\mathsf {T}}}$ is the transpose of ${\displaystyle \mathbf {x} ,}$ ${\displaystyle \mathbf {z} ^{*}}$ is the conjugate transpose of ${\displaystyle \mathbf {z} ,}$ and ${\displaystyle \mathbf {0} }$ denotes the n dimensional zero-vector.

An ${\displaystyle n\times n}$ symmetric real matrix ${\displaystyle M}$ is said to be positive-definite if ${\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} >0}$ for all non-zero ${\displaystyle \mathbf {x} }$ in ${\displaystyle \mathbb {R} ^{n}.}$ Formally,

${\displaystyle M{\text{ positive-definite}}\quad \iff \quad \mathbf {x} ^{\mathsf {T}}M\mathbf {x} >0{\text{ for all }}\mathbf {x} \in \mathbb {R} ^{n}\setminus \{\mathbf {0} \}}$

An ${\displaystyle n\times n}$ symmetric real matrix ${\displaystyle M}$ is said to be positive-semidefinite or non-negative-definite if ${\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} \geq 0}$ for all ${\displaystyle \mathbf {x} }$ in ${\displaystyle \mathbb {R} ^{n}.}$ Formally,

${\displaystyle M{\text{ positive semi-definite}}\quad \iff \quad \mathbf {x} ^{\mathsf {T}}M\mathbf {x} \geq 0{\text{ for all }}\mathbf {x} \in \mathbb {R} ^{n}}$