SPECIAL MATRICES

Symmetric & Skew Symmetric Matrices:

A square matrix A = [aij] is said to be symmetric when aij = aji for all i and j. 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.

For example : $\large  \left[ \begin{array}{ccc} a & h & g    \\ h & b & f  \\ g & f  & c   \end{array} \right]$ is a Symmetric Matrix and $\large   \left[ \begin{array}{ccc} 0 & h & -g    \\ -h & 0 & f  \\ g & -f & 0   \end{array} \right]$ is a Skew-Symmetric Matrix .

Singular and Non-singular Matrix:

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 determinant of square matrix A e.g.

$\large A =  \left[ \begin{array}{cc} 2  & 3     \\ 4 & 5     \end{array} \right]$

$\large |A | =  \left| \begin{array}{cc} 2  & 3     \\ 4 & 5     \end{array} \right|$

= 10 – 12 = -2

⇒  A is a non-singular matrix.

Orthogonal Matrix:

Any square matrix A of order n is said to be orthogonal if AA’ = A’ A = In

Idempotent Matrix:

A square matrix A is called idempotent provided it satisfies the relation A2 = A.
For example: The matrix      $\large A =  \left[ \begin{array}{ccc} 2  & -2 & -4    \\ -1 & 3 & 4 \\ 1 & -2 & -3     \end{array} \right]$   is idempotent as

$\large A^2 = A.A =  \left[ \begin{array}{ccc} 2  & -2 & -4    \\ -1 & 3 & 4 \\ 1 & -2 & -3     \end{array} \right] \left[ \begin{array}{ccc} 2  & -2 & -4    \\ -1 & 3 & 4 \\ 1 & -2 & -3     \end{array} \right]$

$\large  =  \left[ \begin{array}{ccc} 2  & -2 & -4    \\ -1 & 3 & 4 \\ 1 & -2 & -3     \end{array} \right] = A $

Involutary Matrix:

A matrix such that A2 = I is called involutary matrix.

Nilpotent Matrix:

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

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