1. Classification of Matrices, Definitions and its Important Properties

This article will cover about the classification of Matrices ,Definitions of various matrices and its properties .

Matrix: Systematic arrangement of M x N numbers in the form of M Rows and N Columns is said to be a matrix.

Notation:

Principle Diagram Elements:-

The element of a square matrix A such that i = j are said to be the principle diagonal elements.

Various types of Matrices which are useful for GATE Exam are explained below with examples

a.Upper Traingular Matrix:

a square matrix is said to be Upper Triangular Matrix if all the entries below the Main diagonal are zero.

B.Lower Triangular Matrix

A square matrix is said to be Lower triangular Matrix if all the entries above the main diagonal are zero.

C.Diagonal Matrix

For a square Matrix if all the elements except principal diagonal elements are zero then the matrix is known as Diagonal Matrix.

D.Scalar Matrix

The square matrix is said to be scalar matrix if it is having a constant value for all the elements of the principal diagonal and the other elements of the matrix are zero.

Note:

i. If all the elements of a Scalar Matrix are 1 then the matrix is said to be Identity Matrix or Unit matrix and it is denoted by (I_n)

ii. Constant X [Identity Matrix] = [Scalar Matrix]

E.Complex Matrix

If at least one of element of Matrix is Complex Number then the matrix is said to be Complex Matrix.

F.Conjugate Matrix

If each element of a Matrix is replaced with its Complex Conjugate Number then this matrix is said to be Conjugate matrix of the given matrix.

G.Hermition Matrix

If the Transpose of a congugate matrix is equal to the given matrix then it is said to be Hermition Matrix i.e.

H.Skew – Hermition Matrix

If the Transpose of a Congugate matrix is equal to the given matrix but with opposit sign, then the matrix is said to be Skew- Hermition Matrix i.e.

Note:

If A is Hermition matrix then, iA is Skew – Hermition Matrix and Vice – Versa.

I.Orthogonal Matrix

If the Inverse of matrix is equal to its Transpose, then the particular matix is said to be Orthogonal Marix i.e.

Note:

If AB = 0 then A (or) B need not be null matrix

J.Null Matrix:

If all the elements of a matrix are zero, then the matrix is said to be a Null Matrix.

So far we have seen different types of Matrices and now we will discuss some important Properties of Matrices.

PROPERTIES OF A MATRIX

a. Trace of a Matrix:

Trace of a matrix is defined only for square matrix and it is given as the sum of the elelments in principal diagonal.

Trace of a Matrix A is denoted as Tr(A) i.e.

Properties of Trace of a Matrix:

b. Traspose of a Matrix:

The transpose of a given matrix A is obtained by changing the rows into columns and columns into rows . It is denoted as A^T

That means

Properties of Transpose of a Matrix:

Next article will cover the topics about Inverse of a Matrix,Co-factor of a Matrix,Adjoint of a Matrix,Minor of a Matrix and Rank of a Matrix