Special Matrices

Symmetric and Skew Matrices

A square matrix A = [a_ij] is said to be symmetric when a_ij = a_ij for all i and j. If a_ij = -a_ij for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix.

For example:

is a symmetric matrix and is a skew-symmetric matrix.

Hermitian and Skew - Hermitian Matrices

A square matrix A = [a_ij] is said to be Hermitian matrix if a_ij = _ij, i, j i.e. A^θ = -A.

For example:

        are Hermitian matrices.

Note:   
 * If A is a Hermitian matrix then a_ii =_ij is a real i. Thus every   diagonal element of a Hermitian Matrix must be real.
 * A Hermitian matrix over the set of real numbers is actually a real symmetric matrix.

And a square matrix, A = [a_ij] is said to be a skew-Hermitian if a_ij =
-_ij, i, j i.e. A^θ = -A.

For example:

        are skew-Hermitian matrices. 

Note:
 * If A is a skew-Hermitian matrix then a_ii =_ij => a_ii + _ii = 0, i.e. a_ii must be purely imaginary or zero.
 * A skew-Hermitian Matrix over the set of real numbers is actually a real skew- symmetric matrix.

Singular and Non-singular Matrices

Any square matrix A is said to be non-singular if |A| ≠ 0, and a square matrix A is said to be singular if |A| = 0. Here |A| (or det(A) or simply det A) means corresponding determinants of square matrix A e.g. if

10 - 12 = -2 => A is a non-singular matrix.