Definite matrix

In mathematics, a symmetric matrix   M   {\displaystyle \ M\ } with real entries is positive-definite if the real number   x ⊤ M x   {\displaystyle \ \mathbf {x} ^{\top }M\mathbf {x} \ } is positive for every nonzero real column vector   x   , {\displaystyle \ \mathbf {x} \ ,} where   x ⊤   {\displaystyle \ \mathbf {x} ^{\top }\ } is the row vector transpose of   x   . {\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   z ∗ M z   {\displaystyle \ \mathbf {z} ^{*}M\mathbf {z} \ } is positive for every nonzero complex column vector   z   , {\displaystyle \ \mathbf {z} \ ,} where   z ∗   {\displaystyle \ \mathbf {z} ^{*}\ } denotes the conjugate transpose of   z   . {\displaystyle \ \mathbf {z} ~.}