1. Presentation on Matrices and some special matrices In partial fulfillment of the subject Vector calculus and linear algebra (2110015) Submitted by: Agarwal Ritika (130120116001) /IT/C-1 Akabari Nirali (130120116002) /IT/C-1 Akanksha Sharma (130120116003) /IT/C-1 GANDHINAGAR INSTITUTE OF TECHNOLOGY

3. 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

4. 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.

5. S OME 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 rows eg:

6. 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:

7. 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:

8. 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 =

9. 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.

10. Some special matrices: Symmetric matrix: A square matrix A that is equal to its transpose, i.e., A = A T or is a symmetric matrix Skew symmetric matrix: A was equal to the negative of its transpose, i.e., A =− A T, then A is a skew symmetric matrix eg:

11. 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

12. 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

13. Unitary matrix: A square matrix is called unitary if AA*= A*A= I Orthogonal matrix: A square matrix A is called orthogonal if A T A=AA T =I.