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Matrices: Definition 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 = [aij], 1 < i < m,
1 < j < n, where is, j ε i or simply [aij] m × n.
The numbers a11, a12, ... etc of this rectangular array are called the elements of the matrix. The element aij belongs to the ith row and the jth column and is called the (i, j)th element of the matrix.
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 MATRICES
A matrix having a single row is called a row matrix. e.g. [1 3, 5, 7]
A matrix having a single column is called a column matrix. e.g..
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. tr(A). Thus if A = [aij]n×n,
then tr(A) = ∑ni=1 aii = a11 + a22 +......+ ann.
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 = [aij]n×n to be a diagonal matrix, aij = 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).
A diagonal matrix whose all the elements are equal is called a scalar matrix.
For a square matrix A = [aij]n×n to be a scalar matrix, aij =, 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 In.
Thus a square matrix A = [aij]n×n is a unit matrix if aij =.
For example :
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 = [aij]n×n,
for upper triangular matrix, aij = 0, i > j
and for lower triangular matrix, aij = 0, i < j.
* Diagonal matrix is both upper and lower triangular
* A triangular matrix A = [aij]n×n is called strictly triangular if aii = 0 for 1 < i < n.
For example :
are respectively upper and lower triangular matrices.
If all the elements of a matrix (square or rectangular) are zero, it is called a null or zero matrix.
For A = [aij] to be null matrix, aij = 0 i, j.
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 = [aij]m×n and A' = [bij]n×m then bij = aij, i, j.
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 .
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 Aq. The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e.
If A = [aij]m×n, then AΘ = [bij]n×m where bji =
i.e. the (j, i)th element of AΘ = the conjugate of (i, j)th element of A.
If A = then AΘ = .
Properties of Transpose conjugate
(i) (AΘ)Θ = A
(ii) (A + B)Θ = AΘ + BΘ
(iii) (kA)Θ = AΘ, k being any number
(iv) (AB)Θ = BΘAΘ
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