# Positive-definite matrix

algebra concept

A positive-definite matrix is a matrix with special properties. The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices.

## Definition

A square matrix filled with real numbers is positive definite if it can be multiplied by any non-zero vector and its transpose and be greater than zero. The vector chosen must be filled with real numbers.

## Examples

• The matrix ${\displaystyle M_{0}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}$  is positive definite. To prove this, we choose a vector with entries ${\displaystyle {\textbf {z}}={\begin{bmatrix}z_{0}\\z_{1}\end{bmatrix}}}$ . When we multiply the vector, its transpose, and the matrix, we get: ${\displaystyle {\begin{bmatrix}z_{0}&z_{1}\end{bmatrix}}{\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\begin{bmatrix}z_{0}\\z_{1}\end{bmatrix}}={\begin{bmatrix}z_{0}\cdot 1+z_{1}\cdot 0&z_{0}\cdot 0+z_{1}\cdot 1\end{bmatrix}}{\begin{bmatrix}z_{0}\\z_{1}\end{bmatrix}}=z_{0}^{2}+z_{1}^{2};}$

when the entries z0, z1 are real and at least one of them nonzero, this is positive. This proves that the matrix ${\displaystyle M_{0}}$  is positive-definite.