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Definite matrix

Property of a mathematical matrixIn mathematics, a symmetric matrixM{\displaystyle M} with real entries is positive-definite if the real number xTMx{\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} } is positive for every nonzero real column vectorx,{\displaystyle \mathbf {x} ,} where xT{\displaystyle \mathbf {x} ^{\mathsf {T}}} is the row vectortranspose of x.{\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 z∗Mz{\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} .}Positive semi-definite matrices are defined similarly, except that the scalars xTMx{\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} } and z∗Mz{\displaystyle \mathbf {z} ^{*}M\mathbf {z} } are required to be positive or zero (that is, nonnegative).

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