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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.