Presentationon

Matrices and some special matrices

In partial fulfillment of the subjectVector calculus and linear algebra

(2110015)Submitted by:Agarwal Ritika (130120116001) /IT/C-1Akabari Nirali (130120116002) /IT/C-1Akanksha Sharma (130120116003) /IT/C-1

GANDHINAGAR INSTITUTE OF TECHNOLOGY

Matrices and some

special matrices

INTRODUCTION: A matrix is a rectangular table of elements which

may be numbers or abstract quantities that can be added and multiplied .

Matrices are used to describe linear equations, record data that depends on multiple parameters.

There are many applications of matrices in maths viz. graph theory, probality theory, statistics , computer graphics, geometrical optics,etc

Matrix: A set of mn elements arranged in a

rectangular array of m rows and n columns is called a matrix of order m by n, written as m*n.

SOME DEFINITIONS ASSOCIATED WITH MATRICES:

Row matrix:A matrix having only one row and any number of columns

eg:

Column matrix:

A matrix having one column and any number of rowseg:

Zero or null matrix:A matrix whose all the elements are zero is called zero matrix

eg:

Diagonal matrix:A square matrix all of whose non-diagonal elements are zero and at least one diagonal elements is non-zero

eg:

Unit or identity matrix:A diagonal matrix all of whose diagonal elements are unity is called a unit or identity matrix and is denoted by I

eg:

Scalar matrix: A diagonal matrix all of whose diagonal elements are equal is called a scalar matrix

eg:

Upper triangular matrix:A square matrix in which all the elements below the diagonal are zero is called upper triangular matrix

eg:

Lower triangular matrix:A square matrix in which all the elements above the diagonal are zero is called a lower triangular matrix

eg:

Trace of a matrix:The sum of all diagonal elements of a square matrix

eg: trace of A =1+4+6+11

A =

Transpose of a matrix:a matrix obtained by interchanging rows and columns of a matrix is called transpose of a matrix and is denoted by A’

eg:

Determinant of a matrix:if A is a square matrix then determinant of A is represented as IAI or det(A)

Singular and non singular matrices:a square matrix A is called singular if det(A) =0 and non-singular if det(A)≠0.

Some special matrices: Symmetric matrix:

A square matrix A that is equal to its transpose, i.e., A = AT or is a  symmetric matrix

Skew symmetric matrix: A was equal to the negative of its transpose, i.e., A =−AT, then A is a skew symmetric matrix

eg:

Conjugate of a matrix:A matrix obtained from any given matrix A, on replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A and is denoted by A

Transposed conjugate of a matrix:The conjugate of the transpose of a matrix A is called the transposed conjugate or conjugate transpose of A and is denoted by   

Hermitian matrix:A square matrix is called Hermitian if

eg:

Skew Hermitian matrix: A square matrix is called skew matrix if A = −A*

eg: if then

 

Unitary matrix:A square matrix is called unitary if AA*= A*A=I

Orthogonal matrix:A square matrix A is called orthogonal if AT A=AAT =I.