# Types of matrices

Definition: A set of mn numbers (real or imaginary) arranged in the form of a rectangular array of m rows and n columns is called an m x n matrix (to be read as ‘m’ by ‘n’ matrix). An m x n matrix is usually written as $$A=\left[ \begin{matrix}{{a}_{11}} & {{a}_{12}} & {{a}_{13}} & \ldots & {{a}_{1j}} & \ldots & {{a}_{1n}} \\{{a}_{21}} & {{a}_{22}} & {{a}_{23}} & \ldots & {{a}_{2j}} & \ldots & {{a}_{2n}} \\\vdots & {} & {} & {} & {} & {} & \vdots \\{{a}_{i1}} & {{a}_{i2}} & {{a}_{i3}} & \ldots & {{a}_{ij}} & \ldots & {{a}_{im}} \\\vdots & \vdots & \vdots & {} & \vdots & {} & \vdots \\{{a}_{m1}} & {{a}_{m2}} & {{a}_{m3}} & \ldots & {{a}_{mj}} & \ldots & {{a}_{mm}} \\\end{matrix} \right]$$.

In compact form the above matrix is represented by A = [aij] mxn or, A = [aij].

TYPES OF MATRICES:

Row Matrix: A matrix having only one row is called a row-matrix or a row-vector. For example, A = [1 2 -1 -2] is a row matrix of order 1 x 4.

Column Matrix: A matrix having only one column is called a column matrix or a column-vector.

For example, $$A=\left[ \begin{matrix}1 \\ 2 \\-1 \\\end{matrix} \right]$$ and $$B=\left[ \begin{matrix}3 \\2 \\5 \\4 \\\end{matrix} \right]$$ are column-matrices of order 3 x 1 and 4 x 1 respectively.

Square Matrix: A matrix in which the number of rows is equal to the number of columns, say n, is called a square matrix of order n.

Diagonal Matrix: A square matrix A = [aij] nxn is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero i.e. aij = 0, for all i ≠ j

A diagonal matrix of order n x n having d1, d2, …, dn as diagonal elements is denoted by diag [d1, d2, …, dn].

For example, the matrix $$A=\left[ \begin{matrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3 \\\end{matrix} \right]$$ is a diagonal matrix, to be denoted by A = diag [1, 2, 3].

Scalar Matrix: A square matrix A = [aij] nxn is called a scalar matrix if

1. aij = 0 for all i ≠ j , and

2. aij = c for all i, where c ≠ O.

For example, the matrices $$A=\left[ \begin{matrix}2 & 0 \\0 & 2 \\\end{matrix} \right]$$ and $$B=\left[ \begin{matrix}1-2i & 0 & 0 \\ 0 & 1-2i & 0 \\0 & 0 & 1-2i \\\end{matrix} \right]$$

Are scalar matrices of order 2 and 3 respectively.

Identity OR Unit Matrix: A square matrix A = [aij] nxn is called an identity or unit matrix if

1. aij= 0 for all i ≠ j &

2. aij = 1 for all i

For example, the matrices $${{I}_{2}}=\left[ \begin{matrix}1 & 0 \\0 & 1 \\\end{matrix} \right]$$, $$\,{{I}_{3}}=\left[ \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\\end{matrix} \right]$$ are Identity matrices of order 2 and 3 respectively.

Null Matrix: A matrix whose all elements are zero is called a null matrix or a zero matrix.

For example, are null matrices of order 2 x 2 and 2 x 3 respectively.

SPECIAL TYPES OF SQUARE MATRICES

Symmetric Matrix: A square matrix A = [aij] is called a symmetric matrix if aij = aji for all i, j.

For example, the matrix $$A=\left[ \begin{matrix}3 & -1 & 1 \\-1 & 2 & 5 \\ 1 & 5 & -2 \\\end{matrix} \right]$$ is symmetric.

Skew – Symmetric Matrix: A square matrix A = [aij] is a skew-symmetric matrix of aij = – aji for all i, j.

For example, the matrix $$A=\left[ \begin{matrix}0 & 2 & -3 \\-2 & 0 & 5 \\3 & -5 & 0 \\\end{matrix} \right]$$ is a skew – symmetric.

Orthogonal Matrix: A square matrix A is called an orthogonal matrix. If AAT = I = AT A.

Upper Triangular Matrix: A square matrix A = [aij] is called upper triangular matrix if aij = 0 for all i > j.

Thus, in an upper triangular matrix, all elements below the main diagonal are zero.

For example, $$A=\left[ \begin{matrix}1 & 2 & 4 & 3 \\0 & 5 & 1 & 3 \\0 & 0 & 2 & 9 \\0 & 0 & 0 & 5 \\\end{matrix} \right]$$ is an upper triangular matrix.

Lower Triangular Matrix: A square matrix A = [aij] is called a lower triangular matrix if aij = 0 for all i < j.