The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain.

Let L denote the set of n x n positive definite, Hermitian matrices over the complex numbers.

(In fact, every inner product on C arises in this fashion from a Hermitian positive definite matrix.)

For this reason, positive definite matrices play an important role in optimization problems.

Thus the exponential map is a diffeomorphism from onto the space of positive definite matrices.

Let V be a (fixed) positive definite matrix of size p x p.

For instance, systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition.

So, the determinant of a positive definite matrix is less than or equal to the product of its diagonal entries.

Let be defined on , where is a real, positive definite matrix.

Since the matrix is a symmetric positive definite matrix, can be solved twice as fast with the Cholesky decomposition.