Mathematics-I J.Baskar Babujee Department of Mathematics Anna University, Chennai-600 025. MA6151
Content 1.1 INTRODUCTION 1.2 TYPES OF MATRICES 1.3 CHARACTERISTIC EQUATION 1.4 EIGEN VECTORS 1.5 PROBLEMS 1.6 CAYLEY HAMILTON THEOREM
1.7 DIAGONALISATION OF A MATRIX 1.8 REDUCTION OF A MATRIX TO DIAGONAL FORM 1.9 ORTHOGONAL TRANSFORMATION OF A SYMMETRIC MATRIX TO DIAGONAL FORM 1.10 QUADRATIC FORMS 1.11 NATURE OF A QUADRATIC FORM
1.12 REDUCTION OF QUADRATIC FORM TO CANONICAL FORM 1.13 INDEX AND SIGNATURE OF THE QUADRATIC FORM 1.14 LINEAR TRANSFORMATION OF A QUADRATIC FORM. 1.15 REDUCTION OF QUADRATIC FORM TO CANANONICAL FORM BY ORTHOGONAL TRANSFORMATION
Unit 1 MATRICES 1.1 INTRODUCTION:- A matrix is defined as a rectangular array (or arrangement in rows or columns) of scalar subject to certain rules of operations. If mn numbers (real or complex) or functions are arranged in the columns (vertical lines) then A is called an m n matrix. Each of the mn numbers is called an element of the matrix.
An m n matrix is usually written as A matrix is usually denoted by a single capital letter A, B or C etc.
Thus, an m n matrix A may be written as A = , where i = 1, 2, 3, … , m ; j = 1, 2, 3, … , n In Algebra, the matrices have their largest application in the theory of simultaneous equations and linear transformations.
A X = B may be symbolically represented by the equation Where A = , X = , B =
1.2 TYPES OF MATRICES (1) Real Matrix :-A matrix is said to be real if all its elements are real numbers. E.g., is a real matrix.
(2) Square Matrix:- A matrix in which the number of rows is equal to the number of columns is called a square matrix, otherwise, it is said to be rectangular matrix. i.e., a matrix A = is a square matrix if m = n a rectangular matrix if m n
A square matrix having n rows and n columns is called “ n – rowed square matrix”, is a 3 – rowed square matrix The elements of a square matrix are called its diagonal elements and the diagonal along with these elements are called principal or leading diagonal.
The sum of the diagonal elements of a square matrix is called its trace or spur. Thus, trace of the n rowed square matrix A= is
(3) Row Matrix :- A matrix having only one row and any number of columns, i.e., a matrix of order 1 n is called a row matrix. [2 5 -3 0] is a row matrix. Example:-
(4) Column Matrix:- Amatrix having only one column and any number of rows, ii.e., a matrix of order m 1 is called a column matrix. is a column matrix. Example:-
(5) Null Matrix:- A matrix in which each element is zero is called a null matrix or void matrix or zero matrix. A null matrix of order m n is denoted by = Example :-
Example:- (6) Sub - matrix :- A matrix obtained from a given matrix A by deleting some of its rows or columns or both is called a sub – matrix of A. is a sub – matrix of
(7) Diagonal Matrix :- A square matrix in which all non – diagonal elements are zero is called a diagonal matrix. i.e., A = [a ] is a diagonal matrix if a = 0 for i j. is a diagonal matrix. Example:-
(8) Scalar Matrix:- A diagonal matrix in which all the diagonal elements are equal to a scalar, say k, is called a scalar matrix. i.e., A = [a ] is a scalar matrix if is a scalar matrix. Example :-
(9) Unit Matrix or Identity Matrix:- A scalar matrix in which each diagonal element is 1 is called a unit or identity matrix. It is denoted by . i.e., A = [a ] is a unit matrix if is a unit matrix. Example
(10) Upper Triangular Matrix. A square matrix in which all the elements below the principal diagonal are zero is called an upper triangular matrix. i.e., A = [a ] is an upper triangular matrix if a = 0 for i > j is an upper triangular matrix Example:-
- é ù 1 0 0 ê ú 5 6 0 ê ú ê ú 3 2 0 ë û (11) Lower Triangular Matrix. A square matrix in which all the elements above the principal diagonal are zero is called a lower triangular matrix. i.e., A = [a ] is a lower triangular matrix if a = 0 for i < j is a lower triangular matrix. Example:-
(12) Triangular Matrix:- A triangular matrix is either upper triangular or lower triangular. (13) Single Element Matrix:- A matrix having only one element is called a single element matrix. i.e., any matrix  is a single element matrix.
(14) Equal Matrices:- Two matrices A and B are said to be equal iff they have the same order and their corresponding elements are equal. i.e., if A = and B = , then A = B iff a) m = p and n = q b) a = b for all i and j.
Example :- (15) Singular and Non – Singular Matrices:- A square matrix A is said to be singular if |A| = 0 and non – singular if |A| 0. A = is a singular matrix since |A| = 0.
1.3 CHARACTERISTIC EQUATION If A is any square matrix of order n, we can form the matrix , where is the nth order unit matrix. The determinant of this matrix equated to zero, i.e.,
is called the characteristic equation of A. On expanding the determinant, we get where k’s are expressible in terms of the elements a The roots of this equation are called Characteristic roots or latent roots or eigen values of the matrix A.
1.4 EIGEN VECTORS Consider the linear transformation Y = AX ...(1) which transforms the column vector X into the column vector Y. We often required to find those vectors X which transform into scalar multiples of themselves. Let X be such a vector which transforms into X by the transformation (1).
Then Y = X ... (2) From (1) and (2), AX = X AX- X = 0 (A - )X = 0 ...(3) This matrix equation gives n homogeneous linear equations ... (4)
These equations will have a non-trivial solution only if the co-efficient matrix A - is singular i.e., if |A - | = 0 ... (5) Corresponding to each root of (5), the homogeneous system (3) has a non-zero solution X = is called an eigen vector or latent vector
Properties of Eigen Values:- • The sum of the eigen values of a matrix is the sum of the elements of the principal diagonal. • The product of the eigen values of a matrix A is equal to its determinant. • If is an eigen value of a matrix A, then 1/ is the eigen value of A-1 . • If is an eigen value of an orthogonal matrix, then 1/ is also its eigen value.
PROPERTY 1:- If λ1, λ2,…, λn are the eigen values of A, then • k λ1, k λ2,…,k λn are the eigen values of the matrix kA, where k is a non – zero scalar. ii. are the eigen values of the inverse matrix A-1. iii. are the eigen values of Ap, where p is any positive integer.
Proof:- i. Let λr be an eigen value of A and Xr the corresponding eigen vector. Then, by definition, Multiplying both sides by k, (kA)Xr = (kλr)Xr Then k λr is an eigen value of kA and the corresponding eigen vector is the same as that of λr, namely Xr. AXr = λrXr
ii. Pre multiplying both sides of AXr = λrXr by A-1 A-1 (A Xr) = A-1 (λr Xr ) Xr = λr (A-1 Xr ) => A-1 Xr = (Xr) Hence is an eigen value of A-1 and the corresponding eigen vector is the same as that of λr , namely Xr.
iii. Pre multiplying both sides of AXr = λrXr by A A(A Xr) = A(λr Xr ) A2 Xr = λr (AXr ) = λr (λr xr) = λr2 Xr. Similarly, we can prove that A3Xr = λr3 Xr, …, ApXr= λrp Xr, where p is any positive integer. Hence λrp is any eigen value of Ap and the corresponding eigen vector is the same as that of λr, namely Xr.
THEOREM :- A matrix A is singular if and only if 0 is an eigen value of A. 1.5 PROBLEMS 1. Find the sum and product of the eigen values of the matrix without finding the eigen values.
Solution:- Sum of the eigen values of A = sum of its diagonal elements. = - 2 + 1 + 0 = -1. Product of the eigen values of A = | A | = = 45.
2. Two eigen values of the matrix are equal to 1 each. Find the third eigen value. Solution:- Let a be the third eigen value of A. Since sum of the eigen values = sum of the diagonal elements, 1 + 1 + a = 2 + 3 + 2 a = 5 Therefore, the third eigen value of A is 5.
3. The product of two eigen values of the matrix is 16. Find the third eigen value. Solution:- Let a be the third eigen value of A. Since product of the eigen values = | A | 16a = Therefore, a = 2.
4. Find the sum of the eigen values of the inverse of Solution:- The eigen values of the lower triangular matrix A is 1, -3, 2. Then the eigen values of A-1 are Sum of the eigen values of A-1 = =
5. If , find the eigen values of 3A, A-1 and – 2A-1. Solution:- The eigen values of A are 2, -1, 4. The eigen values of 3A are 3×2, 3×(-1), 3×4 i.e., 6, -3, 12
6. Find the eigen values and eigen vectors of the matrix A = Solution:- The characteristic equation of the given matrix is
the eigen values of A are 6, -1. or Thus, Corresponding to =6, the eigen vectors are given by (A – 6 ) X1 = 0
7. Find the eigen values and eigen vectors of the matrix A = Solution:- The characteristic equation of the given matrix is
Thus, the eigen values of A are -3, -3, 5.
Corresponding to λ = 5, the eigen vectors are given by (A – 5 I )X2 = 0.