Positive definite matrixis a symmetric matrix with all eigenvalues greater than 0. A positive definite matrix in linear algebra is a Hermitian matrix with properties similar to positive real numbers in complex numbers. The linear operator corresponding to a positive definite matrix is a symmetric positive definite bilinear form.

Positive definite matrix properties

• The determinant of a positive definite matrix is always positive;
• A real symmetric matrix A is positive definite if and only if A is identical to the identity matrix;
• If A is a positive definite matrix, then the inverse matrix of A is also a positive definite matrix;
• The sum of two positive definite matrices is a positive definite matrix;
• The product of a positive real number and a positive definite matrix is a positive definite matrix.

Positive definite matrix determination

According to the definition and properties of positive definite matrix, there are two methods to determine the positive definiteness of symmetric matrix A:

• Find all eigenvalues of A: If all eigenvalues of A are positive, then A is positive definite; if all eigenvalues of A are negative, then A is negative definite;
• Calculate the principal minors of A of various orders: If the principal minors of various orders of A are all greater than zero, then A is positive definite; if among the principal minors of various orders of A, the odd-order principal minors are negative and the even-order principal minors are positive, then A is negative definite.

Positive definite matrix applications

The properties of positive definite matrices, such as the existence of a unique LDU decomposition, can be further decomposed into GGT if it is real and symmetric, and the triangular decomposition of the matrix can greatly reduce the computational complexity. The covariance matrix of the sample is a real and symmetric positive definite matrix.

Related words: matrix, eigenvalue