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 [math]\displaystyle{ M_0 = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} }[/math] is positive definite. To prove this, we choose a vector with entries [math]\displaystyle{ \textbf{z}= \begin{bmatrix} z_0 \\ z_1\end{bmatrix} }[/math]. When we multiply the vector, its transpose, and the matrix, we get: [math]\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; }[/math]

when the entries z₀, z₁ are real and at least one of them nonzero, this is positive. This proves that the matrix [math]\displaystyle{ M_0 }[/math] is positive-definite.