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

In compact form the above matrix is represented by

A = [a_ij] _mxn or, A = [a_ij]

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,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 = [a_ij] _nxn is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero i.e.

a_ij = 0, for all i ≠ j

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

For example, the matrixis a diagonal matrix, to be denoted by A = diag [1, 2, 3].

Scalar Matrix

 A square matrix A = [a_ij] _nxn is called a scalar matrix if

i.          a_ij = 0 for all i ≠ j  and

ii.        a_ij = c for all i, where c ≠ 0.

For example, the matrices

Are scalar matrices of order 2 and 3 respectively.

Identity Or Unit Matrix

A square matrix A = [a_ij] _nxn is called an identity or unit matrix if

i.       a_ij= 0 for all i ≠ j and

ii.     a_ij = 1 for all i

For example, the matricesare 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 = [a_ij] is called a symmetric matrix if a_ij = a_ji for all i, j

For example, the matrixis symmetric,

Skew-Symmetric Matrix

 A square matrix A = [a_ij] is a skew-symmetric matrix of a_ij = - a_ji for all i, j

For example, the matrixis a skew-symmetric

Orthogonal Matrix

A square matrix A is called an orthogonal matrix. If

AA^T =I=A^T A

Upper Triangular Matrix

A square matrix A = [a_ij] is called upper triangular matrix if a_ij = 0 for all i >j.

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

For example is an upper triangular matrix.

Lower Triangular Matrix

A square matrix A = [a_ij] is called a lower triangular matrix if a_ij = 0 for all i < j.