Definite matrix

In mathematics, a symmetric matrix ${\displaystyle \ M\ }$ with real entries is positive-definite if the real number ${\displaystyle \ \mathbf {x} ^{\top }M\mathbf {x} \ }$ is positive for every nonzero real column vector ${\displaystyle \ \mathbf {x} \ ,}$ where ${\displaystyle \ \mathbf {x} ^{\top }\ }$ is the row vector transpose of ${\displaystyle \ \mathbf {x} ~.}$^[1] 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 ${\displaystyle \ \mathbf {x} ^{\top }M\mathbf {x} \ }$ and ${\displaystyle \ \mathbf {z} ^{*}M\mathbf {z} \ }$ are required to be positive or zero (that is, not negative). 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.