Matrices lecture one

Definition: A rectangular array of numbers enclosed by a pair of brackets, such as

(a) 

matrix (a)

  and (b)

And subject to certain rules of operation given below is a matrix.

Order: A matrix of m rows and n columns is said to be of order “m by n” or m × n.

Sum of matrix: Two matrixes have to be same order. Two matrices of the same order are said to be conformable for addition or subtraction.

Multiplication: The product AB is defined or A is conformable to B for multiplication only when the number of columns of A is equal to the number of rows of B.

Square matrix: When m=n, (1,1)  is square and will be called a square matrix of

square matrix

order n or an n-square matrix. Such as          

Equal matrices: Two matrices A and B are said to be equal if and only if the have the same order and each element of one is equal to the corresponding element of the other, that is, if and only if

a_ij =b_ij.

Idempotent matrices:  A² = A where A is  a square matrices.

Nilpotent matrices:  A matrix A for which A^p = 0, where p is a positive integer, is called nilpotent.

Inverse matrix: AB=I, then  B is the inverse matrix of A

Periodic matrix: A^k+1 = A where k=least positive integer and k= periodic value

Singular matrix: If determinant = 0 then its called singular matrix. 

Transpose matrix: The matrix of order n×n obtained by interchanging the rows and column of an m×n matrix A is called the transpose of A and is denoted by  A^’ ( A transpose). For example

Conjugate matrix: The  complex number a + bi and a - bi  are called conjugates, each being the conjugate of the  other. If z = a +bi, its conjugate denoted by

      ( only changed of sign of unreal number).

Symmetric Matrices: A = A'   , where A’ is transpose matrix

Skew-symmetric matrix: A = - A'

Hermitian matrix:  A square matrix A = [a_ij] such that

hermitian

is called Hermitian.

skew hermitian

Skey-hermitian matrix: 

Direct Sum: Let A ₁, A₂, ……..As be square matrix of respective orders m₁, m₂, …….m_s.
The generalization A = 

=  dig ( A₁ , A₂ ,……. A_s)

Of the diagonal matrix is called the direct sum of the A_i.

Orthogonal matrix:  AA’ = I

Unitary Matrix

Unitary matrix :    , where

 is transpose conjugates.