Matrices: Definition and Classification

Definition of Materices and Classification

A rectangular array of symbols (which could be real or complex numbers) along rows and columns is called a matrix.

Thus a system m × n symbols arranged in a rectangular formation along m rows and n columns and bounded by the brackets [.] is called an m by n matrix (which is written as m x n matrix)

i.e. A =  is a matrix of order m × n.

In a compact form the above matrix is represented by A = [a_ij], 1 < i < m,
1 < j < n, where is, j ε i or simply [a_ij] m × n.

The numbers a₁₁, a₁₂, ... etc of this rectangular array are called the elements of the matrix. The element a_ij belongs to the i^th row and the j^th column and is called the (i, j)^th element of the matrix.

Equal Matrices

Two matrices are said to be equal if they have the same order and each element of one is equal to the corresponding element of the other.
 

Classification of Matarices

Row Matrix

A matrix having a single row is called a row matrix. e.g. [1 3, 5, 7]

Column Matrix

A matrix having a single column is called a column matrix. e.g_..

Square Matrix

An m x n matrix A is said to be a square matrix if m = n i.e. number of rows = number of columns.

For Example: A = is a square matrix of order 3 × 3.

Note: In a square matrix the diagonal from left hand side upper corner to right hand side lower corner is known as leading diagonal or principal diagonal. In the above example square matrix containing the elements 1, 3, 5 is called the leading or principal diagonal.

Traces of a Matrix

The sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A i.e. t_r(A). Thus if A = [a_ij]_n×n,

then t_r(A) = ∑ⁿ_i=1  a_ii = a₁₁ + a₂₂ +......+ a_nn.

Diagonal Matrix

A square matrix all of whose elements except those in the leading diagonal, are zero is called a diagonal matrix. For a square matrix A = [a_ij]_n×n to be a diagonal matrix, a_ij = 0, whenever i ≠ j.

For example: A = is a diagonal matrix of order 3 × 3.

Note:  Here A can also be represented as diag (3,5 -1).

Scalar Matrix

A diagonal matrix whose all the elements are equal is called a scalar matrix.

For a square matrix A = [a_ij]_n×n to be a scalar matrix, a_ij =, where m ≠ 0.

For example: A =  is a scalar matrix.

Unit Matrix or Identity Matrix

A diagonal matrix of order n which has unity for all its elements, is called a unit matrix of order n and is denoted by I_n.

Thus a square matrix A = [a_ij]_n×n is a unit matrix if a_ij =.

For example :

I₃ =.

Triangular Matrix

A square matrix in which all the elements below the principal diagonal are zero is called Upper Triangular matrix and a square matrix in which all the elements above the principal diagonal are zero is called Lower Triangular matrix.

Given a square matrix A = [a_ij]_n×n,

for upper triangular matrix, a_ij = 0,       i > j

and for lower triangular matrix, a_ij = 0,   i < j.

Note :

• Diagonal matrix is both upper and lower triangular

• A triangular matrix A = [a_ij]_n×n is called strictly triangular if a_ii = 0 for 1 < i < n.

For Example :

are respectively upper and lower triangular matrices.

 Null Matrix

If all the elements of a matrix (square or rectangular) are zero, it is called a null or zero matrix.

For A = [a_ij] to be null matrix, a_ij = 0 i, j.

For Example:

        is a zero matrix.

Transpose of a Matrix

The matrix obtained from any given matrix A, by interchanging its rows and columns, is called the transpose of A and is denoted by A'.

If A = [a_ij]_m×n and A' = [b_ij]_n×m then b_ij = a_ij, i, j.

For Example:

        If A = , then A' =
 

Properties of Transpose

(i)     (A')' = A

(ii)    (A + B)' = A' + B', A and B being conformable matrices

(iii)   (αA)' = αA', α being scalar

(iv)   (AB)' = B'A', A and B being conformable for multiplication

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 .

For Example:

 A = . Then,   

Properties of Conjugate

​Transpose Conjugate of a Matrix

The transpose of the conjugate of a matrix is called transposed conjugate of A and is denoted by A^q. The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e.

= A^Θ.

If A = [a_ij]_m×n, then A^Θ = [b_ij]_n×m where b_ji =

i.e. the (j, i)^th element of A^Θ = the conjugate of (i, j)^th element of A.

For Example:

     If A =   then A^Θ = .

Properties of Transpose Conjugate

(i)    (A^Θ)^Θ = A

(ii)   (A + B)^Θ = A^Θ + B^Θ

(iii)   (kA)^Θ = A^Θ, k being any number

(iv)   (AB)^Θ = B^ΘA^Θ
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