Positive-definite matrix

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In linear algebra, a positive-definite matrix is a matrix which in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form (or a sesquilinear form in the complex case).

The proper definition of positive-definite is unambiguous for Hermitian matrices, but there is no agreement in the literature on how this should be extended for non-Hermitian matrices, if at all. (See the section Non-Hermitian matrices below.)

Contents

Definition

An n × n real symmetric matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (z \in \mathbb{R}^n), where zT denotes the transpose of z.

An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z. The quantity z*Mz is always real because M is a Hermitian matrix.

Examples

The matrix  M_0 =  \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} is positive definite. For a vector with entries \textbf{z}= \begin{bmatrix} z_0 \\ z_1\end{bmatrix} the quadratic form is  \begin{bmatrix} z_0 & z_1\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} z_0 \\ z_1\end{bmatrix}=z_0^2+z_1^2; when the entries z0, z1 are real and at least one of them nonzero, this is positive.

The matrix  M_1 =  \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} is not positive definite. When \textbf{z}= \begin{bmatrix} 1\\ -1\end{bmatrix} the quadratic form at z is then

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