SPECIAL MATRICES

Symmetric and Skew Symmetric Matrices:

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

Orthogonal Matrix:

Any square matrix A of order n is said to be orthogonal if AA¢ = A¢ A = .

Idempotent Matrix:

A square matrix A is called idempotent provided it satisfies the relation A2 = A.

Involuntary Matrix:

A square matrix A is said to be involuntary if A2 = I.

Nilpotent Matrix:

A square matrix A is called a nilpotent matrix if there exists a positive integer m such that
Am = O, where O is a null matrix. If m is the least positive integer such that Am = O, then m is called the index of the nilpotent matrix A.