An n×n complex matrix A is called positive definite if ℜ[x^* A x]>0 for all nonzero complex vectors x element C^n, where x^* denotes the conjugate transpose of the vector x. In the case of a real matrix A, equation (-1) reduces to x^T A x>0, where x^T denotes the transpose. Positive definite matrices are of both theoretical and computational importance in a wide variety of applications. They are used, for example, in optimization algorithms and in the construction of various linear regression models.

determinant | eigenvalue | Hermitian matrix | matrix | negative definite matrix | negative semidefinite matrix | positive eigenvalued matrix | positive matrix | positive semidefinite matrix | Sylvester's criterion