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 $M_0=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ is positive definite. To prove this, we choose a vector with entries $\mathbf{\text{z}}=\left[\begin{array}{c}{z}_0\\ z_1\end{array}\right]$. When we multiply the vector, its transpose, and the matrix, we get: $\left[\begin{array}{cc}{z}_0& z_1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{z}_0\\ z_1\end{array}\right]=\left[\begin{array}{cc}{z}_0\cdot 1+z_1\cdot 0& z_0\cdot 0+z_1\cdot 1\end{array}\right]\left[\begin{array}{c}{z}_0\\ z_1\end{array}\right]=z_0^2+z_1^2;$

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.