Eigenvalue Algorithm - Eigenvalues and Eigenvectors - Normal, Hermitian, and Real-symmetric Matrices

Normal, Hermitian, and Real-symmetric Matrices

The adjoint M* of a complex matrix M is the transpose of the conjugate of M: M * = M T. A square matrix A is called normal if it commutes with its adjoint: A*A = AA*. It is called hermitian if it is equal to its adjoint: A* = A. All hermitian matrices are normal. If A has only real elements, then the adjoint is just the transpose, and A is hermitian if and only if it is symmetric. When applied to column vectors, the adjoint can be used to define the canonical inner product on C n: w • v = w* v. Normal, hermitian, and real-symmetric matrices have several useful properties:

• Every generalized eigenvector of a normal matrix is an ordinary eigenvector.
• Any normal matrix is similar to a diagonal matrix, since its Jordan normal form is diagonal.
• Eigenvectors of distinct eigenvalues of a normal matrix are orthogonal.
• For any normal matrix A, C n has an orthonormal basis consisting of eigenvectors of A. The corresponding matrix of eigenvectors is unitary.
• The eigenvalues of a hermitian matrix are real, since (λ - λ)v = (A* - A)v = (A - A)v = 0 for a non-zero eigenvector v.
• If A is real, there is an orthonormal basis for R n consisting of eigenvectors of A if and only if A is symmetric.

It is possible for a real or complex matrix to have all real eigenvalues without being hermitian. For example, a real triangular matrix has its eigenvalues along its diagonal, but in general is not symmetric.