Positive Definite Matrices

Examples:

• potential energy near a stable equilibrium;
• kinetic energy;
• finite element computations;
• ...

Hermitian Matrices:

A matrix A is Hermitian if A^H = A.

Hermitian matrices always have real eigenvalues and a complete set of orthonormal eigenvectors. The corresponding transformation matrix is unitary.

Positive Definite Matrices:

A matrix is positive definite if it is Hermitian and

for all nonzero .

Note: not all authors require the matrix to be Hermitian to be positive definite.

One way to check positive definiteness of a matrix is to verify whether all eigenvalues are positive:

10/10/01 0:42:38