Module-2

Eigen values and eigen vectors

2.1 Eigen values and eigen vectors

Characteristic equation: -

Let A he a square matrix, be any scalar then is called characteristic equation of a matrix A.

Note:

Let a be a square matrix and ‘’ be any scalar then,

1)     is called characteristic matrix

2)     is called characteristic polynomial.

The roots of a characteristic equations are known as characteristic root or latent roots, Eigen values or proper values of a matrix A.

Eigen vector:-

Suppose be an Eigen value of a matrix A. Then a non – zero vector x1 such that.

     … (1)

Such a vector ‘x1’ is called as Eigen vector corresponding to the Eigen value .

Properties of Eigen values:-

If

are n Eigen values of square matrix A then

are m Eigen values of a matrix A-1.

Properties of Eigen vector:-

Example1: Find the sum and the product of the Eigen values of ?

Sol.   The sum of Eigen values =  the sum of the diagonal elements

   =1+(-1)=0

     The product of the Eigen values is the determinant of the matrix

       On solving above equations we get

Example2: Find out the Eigen values and Eigen vectors of ?

Sol.  The Characteristics equation is given by

Or

Hence the Eigen values are 0,0 and 3.

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

This implies that

Here number of unknowns are 3 and number of equation is 1.

Hence we have (3-1) = 2 linearly independent solutions.

Let

Thus the Eigen vectors corresponding to the Eigen value are (-1,1,0) and (-2,1,1).

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

This implies that

Taking last two equations we get

Or

Thus the Eigen vectors corresponding to the Eigen value are (3,3,3).

Hence the three Eigen vectors obtained are  (-1,1,0), (-2,1,1) and (3,3,3).

Example3: Find out the Eigen values and Eigen vectors of

Sol:                                          Let A =

The characteristics equation of A is .

                                           Or 

                                                Or 

                                                      Or

                                                       Or

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

         Or

On solving we get 

Thus the Eigen vectors corresponding to the Eigen value is (1,1,1).

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

or     

On solving or .

Thus the Eigen vectors corresponding to the Eigen value is (0,0,2).

The Eigen vector corresponding to Eigen value is

Where X is the column matrix of order 3 i.e.

or     

On solving we get  or .

Thus the Eigen vectors corresponding to the Eigen value is (2,2,2).

Hence three Eigen vectors are (1,1,1), (0,0,2) and (2,2,2).

Example-4:

Determine the Eigen values of Eigen vector of the matrix.

Solution:

Consider the characteristic equation as,

i.e.

i.e.

i.e.

Which is the required characteristic equation.

are the required Eigen values.

Now consider the equation

                              … (1)

Case I:

If   Equation (1) becomes

R1 + R2

Thus

independent variable.

Now rewrite equation as,

Put x3 = t

&

Thus .

Is the eigen vector corresponding to .

Case II:

If equation (1) becomes,

Here

Independent variables

Now rewrite the equations as,

Put

&

.

Is the eigen vector corresponding to .

Case III:

If equation (1) becomes,

Here rank of

independent variable.

Now rewrite the equations as,

Put

Thus .

Is the eigen vector for .

Example-5:

Find the Eigen values of Eigen vector for the matrix.

Solution:

Consider the characteristic equation as

i.e.

i.e.

are the required eigen values.

Now consider the equation

                               … (1)

Case I:

Equation (1) becomes,

Thus and n = 3

3 – 2 = 1 independent variables.

Now rewrite the equations as,

Put

,

I.e.

the Eigen vector for

Case II:

If equation (1) becomes,

Thus

Independent variables.

Now rewrite the equations as,

Put

Is the Eigen vector for

Now

Case III:-

If equation (1) gives,

R1 – R2

Thus

Independent variables

Now

Put

Thus

Is the Eigen vector for

Key takeaways-

4.     Let A he a square matrix, be any scalar then is called characteristic equation of a matrix A.

5.     If two are more Eigen values are identical then the corresponding Eigen vectors may or may not be linearly independent.

6.     The Eigen vectors corresponding to distinct Eigen values of a real symmetric matrix are orthogonal

2.2  Symmetric and skew-symmetric matrices

Symmetric matrix:

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So, if aᵢⱼ denotes the entry in the i-th row and j-th column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero.

Any square matrix is said to be symmetric matrix if its transpose equals to the matrix itself.

For example:

&

Example: check whether the following matrix A is symmetric or not?

A = 

Sol. As we know that if the transpose of the given matrix is same as the matrix itself then the matrix is called symmetric matrix.

So that, first we will find its transpose,

Transpose of matrix A ,

Here,

A =

So that, the matrix A is symmetric.

Example: Show that any square matrix can be expressed as the sum of symmetric matrix and anti- symmetric matrix.

Sol.  Suppose A is any square matrix

Then,

A =

Now,

(A + A’)’ = A’ + A

A+A’ is a symmetric matrix.

Also,

(A - A’)’ = A’ – A

Here A’ – A is an anti – symmetric matrix

So that,

Square matrix = symmetric matrix + anti-symmetric matrix

Example: Let us test whether the given matrices are symmetric or not i.e., we check for,

A =

(1)  A =

Now

  =

A =

Hence the given matric symmetric

Example: let A be a real symmetric matrix whose diagonal entries are all positive real numbers.

Is this true for all of the diagonal entries of the inverse matrix A-1 are also positive? If so, prove it. Otherwise give a counter example

Solution: The statement is false, hence we give a counter example

Let us consider the following 22 matrix

A =

The matrix A satisfies the required conditions, that is A is symmetric and its diagonal entries are positive.

The determinant det(A) = (1)(1)-(2)(2) = -3 and the inverse of A is given by

A-1= =

By the formula for the inverse matrix for 22 matrices.

This shows that the diagonal entries of the inverse matric A-1 are negative.

Skew-symmetric:

A skew-symmetric matrix is a square matrix that is equal to the negative of its own transpose. Anti-symmetric matrices are commonly called as skew-symmetric matrices. 

In other words-

Skew-symmetric matrix-

A square matrix A is said to be skew symmetrix matrix if –

1. A’ = -A,    [ A’ is the transpose of A]

2. All the main diagonal elements will always be zero.

For example-

A =

This is skew symmetric matrix, because transpose of matrix A is equals to negative A.

Example: check whether the following matrix A is symmetric or not?

A =

Sol. This is not a skew symmetric matrix, because the transpose of matrix A is not equals to -A.

-A = A’

Example: Let A and B be nn skew-matrices. Namely AT = -A and BT = -B

(a)  Prove that A+B is skew-symmetric.

(b)  Prove that cA is skew-symmetric for any scalar c.

(c)  Let P be an mn matrix. Prove that PTAP is skew-symmetric.

Solution: (a) (A+B)T = AT + BT = (-A) +(-B) = -(A+B)

Hence A+B is skew symmetric.

(b) (cA)T = c.AT =c(-A) = -cA

Thus, cA is skew-symmetric.

(c)Let P be an mn matrix. Prove that PT AP is skew-symmetric.

Using the properties, we get,

(PT AP)T = PTAT(PT)T = PTATp

= PT (-A) P = - (PT AP)

Thus (PT AP) is skew-symmetric.

Key takeaways-

2.3 Orthogonal matrices, Complex matrices

Orthogonal matrix:

An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field.

Suppose A is a square matrix with real elements and of n x n order and AT is the transpose of A. Then according to the definition, if, AT = A-1 is satisfied, then,

A AT = I

Where ‘I’ is the identity matrix, A-1 is the inverse of matrix A, and ‘n’ denotes the number of rows and columns.

“Any square matrix A is said to be an orthogonal matrix if the product of the matrix A and its transpose is an identity matrix”

 Such that,

A. A’ = I

Matrix × transpose of matrix = identity matrix

Note- if |A| = 1, then we can say that matrix A is proper.

Example:    and   are the form of orthogonal matrices.

Example: prove Q= is an orthogonal matrix

Solution:                                  Given Q =

So,

QT = …..(1)

Now, we have to prove QT = Q-1

Now we find Q-1

Q-1 = … (2)

Now, compare (1) and (2) we get QT = Q-1

Therefore, Q is an orthogonal matrix.

Key takeaways-

2.4 Hermitian and skew matrices

Hermitian matrix:

A square matrix A = is said to be hermitian matrix if every element of A is equal to conjugate complex j-ith element of A.

It means,

For example:

Necessary and sufficient condition for a matrix A to be hermitian –

A = (͞A)’

Skew-Hermitian matrix-

A square matrix A = is said to be hermitian matrix if every element of A is equal to negative conjugate complex j-ith element of A.

Note- All the diagonal elements of a skew hermitian matrix are either zero or pure imaginary.

For example:

The necessary and sufficient condition for a matrix A to be skew hermitian will be as follows-

- A = (͞A)’

Note:   A Hermitian matrix is a generalization of a real symmetric matrix and also every real symmetric matrix is Hermitian.

Similarly a Skew- Hermitian matrix is a generalization of a  Skew symmetric matrix  and also every Skew- symmetric matrix  is Skew–Hermitian.

Theorem: Every square complex   matrix   can   be uniquely   expressed as sum hermitian and skew-hermitian matrix.

Or If A   is given square complex matrix then  is   hermitian and  is skew-hermitian matrices.

Example1: Express the matrix A as sum of hermitian and skew-hermitian matrix where

Let

A =

Therefore

 And

Let

Again

Hence P is a hermitian matrix.

Let 

 Again

Hence Q is a skew- hermitian matrix.

We Check

  P +Q=

Hence proved.

Example2: If A =   then show that

(i) is hermitian matrix.

(ii)   is  skew-hermitian matrix.

Given  

A =

Then

Let

Also

Hence P is a Hermitian matrix.

Let

Also

Hence Q is a   skew-hermitian matrix.

Key takeaways-

A square matrix A =

is said to be hermitian matrix if every element of A is equal to conjugate complex j-ith element of A.

A square matrix A =

is said to be hermitian matrix if every element of A is equal to negative conjugate complex j-ith element of A.

2.5 Unitary matrices and similarity of matrices

Unitary matrix-

A square matrix A is said to be unitary matrix if the product of the transpose of the conjugate of matrix A and matrix itself  is an identity matrix.

Such that,

( ͞A)’ . A = I

For example:

A  =   and its    ( ͞A)’ = 

Then  (͞A)’ . A = I

So that we can say that matrix A is said to be a unitary matrix.

2.6  Diagonalisation of Matrices

Two square matrix and A of same order n are said to be similar if and only if

for some non singular matrix P.

Such transformation of the matrix A into with the help of non singular matrix P is known as similarity transformation.

Similar matrices have the same Eigen values.

If X is an Eigen vector of matrix A then is Eigen vector of the matrix

Reduction to Diagonal Form:

Let A be a square matrix of order n has n linearly independent Eigen vectors Which form the matrix P such that

Where P is called the modal matrix and D is known as spectral matrix.

Procedure: let A be a square   matrix of order 3.

Let three Eigen vectors of A are  Corresponding to Eigen values

Let

  {by characteristics equation of A}

Or

Or 

Note: The method of diagonalization is helpful in calculating power of a matrix.

  .Then for an integer n we have  

We are using the example of 1.6*

Example1:  Diagonalise the matrix

Let

A=

The three Eigen vectors obtained are  (-1,1,0), (-1,0,1) and (3,3,3) corresponding to Eigen values .

Then

And

Also we know that    

Example2: Diagonalise the matrix

Let A =

The Eigen vectors are (4,1),(1,-1) corresponding to Eigen values .

Then    and also 

Also we know that    

Key takeaways-

Two square matrix

and A of same order n are said to be similar if and only if

for some non singular matrix P.

2.     If X is an Eigen vector of matrix A then is Eigen vector of the matrix

References