## What are the Special Types of Matrices?

### 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*.

### Hermitian and Skew-hermitian matrix :

A square matrix is said to be hermitian matrix if

### Orthogonal matrix :

A square matrix *A* is called orthogonal if AA^{T} = I = A^{T }A *i.e.,* if A^{−1} = A.

In fact every unit matrix is orthogonal. Determinant of orthonogal matrix is – 1 or 1.

### Idempotent matrix :

A square matrix A is called an idempotent matrix if A^{2} = A.

In fact every unit matrix is indempotent.

### Involutory matrix :

A square matrix A is called an involutory matrix if A^{2} = I or A^{−1} = A.

In fact every unit matrix is involutory.

### Nilpotent matrix :

A square matrix A is called a nilpotent matrix if there exists a p ∈ N such that A^{p} = 0.

Determinant of every nilpotent matrix is 0.

### Periodic matrix :

A matrix A will be called a periodic matrix if where k is a positive integer. If A^{k+1} = A however k is the least positive integer for A^{k+1} = A, then k is said to be the period of A.

Differentiation of a matrix : If then is a differentiation of matrix A.

### Conjugate of a matrix :

The matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex numbers is called conjugate of A and is denoted by \(\overline { A }\).

**Properties of conjugates**

### Transpose conjugate of a matrix :

The transpose of the conjugate of a matrix A is called transposed conjugate of A and is denoted by A^{θ}. The conjugate of the transpose of A is the same as the transpose of the conjugate of A

**Properties of transpose conjugate**