Positive-definite matrix

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

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