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In the case of a complex matrix A, the Hermitian transpose is the complex conjugate transpose of A:
We are ready to give some more definitions of special kinds of matrices:
Normal:
Hermitian:
Skew-Hermitian:
Symmetric (real) matrix:
An example of a Hermitian (and hence normal matrix) is:
5.6.3 Unitary and Orthogonal Matrices
A matrix U is unitary if its inverse equals its Hermitian transpose:
Unitary matrices are normal:
A real square matrix Q is orthogonal if:
(5.17)
Orthogonal matrices are important in numerical linear algebra applications because of their numeric stability properties. Some application areas are matrix decomposition methods (Golub and van Loan (1996)) such as:
QR decomposition orthogonal, upper triangular.
Singular Value Decomposition (SVD) and V orthogonal, diagonal matrix.
Eigendecomposition symmetric, Q orthogonal, diagonal.
Polar decomposition where is orthogonal, symmetric positive-semidefinite.
A particular application area is solving overdetermined systems of linear equations and solving ill-posed linear systems.
5.6.4 Positive Definite Matrices
This is another very important class of matrices. Matrices having this property are highly desirable in applications and algorithms. An
(5.18)
and positive semidefinite if:
(5.19)
Some necessary conditions for positive semidefiniteness are:
The diagonal elements of A must be positive.
A is positive definite if and only if all its eigenvalues are positive.
The element of A having the greatest absolute value must be on the diagonal of A.
An example of a positive definite matrix A is:
To prove this let
We can take the square root
where L is a lower triangular matrix having positive values on its diagonal. Equation (5.20) is called the Cholesky decomposition for A.
5.6.5 Non-Negative Matrices
A matrix A is non-negative (written
