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Eigenvectors and Eigenvalues

During linear transformation there may be certain vectors which remain on its span after transformation and are only enlarged or reduced in size (magnitude is changed and orientation remain same).

Such vectors are called eigenvectors and the corresponding value through which magnitude changes is called eigenvalue.

Suppose a vector v when transformed with matrix A remains on its span and is enlarged by a value λ. Then,

Av = λv           ................ (i)

Eigenvectors and eigenvalues
Eigenvectors and eigenvalues - Physical Significance

Learn Basics: Matrix as linear transformation of Vectors

For example if A and v are as follows:

Eigenvectors and eigenvalues - Physical Significance
Eigenvectors and eigenvalues - Physical Significance

Vector 3j remains on y axis even after transformation and only its magnitude changes by 2. Similarly all vectors on y-axis retain its orientation and are only enlarged by a value 2. Hence, x = 0 is eigenvector with corresponding eigenvalue as 2.

It may be observed that in equation (i), left side is a matrix and right side is vector. To make both side similar, RHS is multiplied by identity matrix.

Av = λIv

(A - λI)v = 0

For a non-zero vector v, this is possible only if

|A - λI| = 0      ............. (ii)

This equation gives eigenvalues and corresponding eigenvectors may be found out by putting value of λ in equation (ii).

Notes:

  1. A square matrix A and its transpose have same eigenvalues.

  2. Sum of eigenvalues is equal to sum of diagonal elements of matrix (trace of a matrix).

  3. Product of eigenvalues is equal to determinant of matrix.

  4. If eigenvalue of A is λ, eigenvalue of (A)^m will be (λ)^m (for inverse of A, it will be 1/λ).

  5. If two or more eigenvalues are equal, there may or may not be linearly independent eigenvectors corresponding to this value.

  6. Two eigenvectors are orthogonal if dot product of vectors is zero.

  7. For a symmetric matrix having two distinct eigenvalues, corresponding eigenvectors are orthogonal.

  8. For a column where element aii ≠ 0 and all other elements aij = 0, one of the eigenvector is that column vector and corresponding eigenvalue is equal to magnitude of the column vector. For example for a 3x3 matrix as given below:

Eigenvectors and eigenvalues

In this case for column 1, a11 = 2 and all other elements are zero. Similarly for column 3, a33 = 3 and all other elements are zero.

Two eigenvalues of matrix are 2 and 3 and corresponding eigenvectors are x axis and z axis.

Normalized Eigenvector

Normalized eigenvector is found out by dividing each element by magnitude of column vector.

Example 1

GATE 2019 (1 Mark): Consider the matrix P. The number of distinct eigenvalues of P is

A) 0

B) 1

C) 2

D) 3

GATE previous year questiond
Consider the matrix P. The number of distinct eigenvalues of P is

Solution: Characteristics equation of the matrix P is

|P - λI| = 0

Consider the matrix P. The number of distinct eigenvalues of P is

(1-λ)[(1-λ)(1-λ) - 0] = 0

λ = 1,1,1

So the equation has only one distinct eigenvalue.

Option B is the correct answer.

Example 2:

GATE 2019 (1 Mark): In matrix equation [A]{X} = {R}, one of the eigenvalues of matrix A is

In matrix equation [A]{X} = {R}, one of the eigenvalues of matrix A is

A) 4

B) 8

C) 15

D) 16

Solution: Vector R may be written as

R = 32i + 16j + 64k = 16 (2i + j + 4k) = 16X

So vector X after transformation remains on its span and only its magnitude is enlarged by 16. Hence, one of eigenvalue of matrix A is 16.

Hence option D is the correct option.

Example 3:

GATE 2017 (1 Mark): Product of eigenvalues of matrix P is

GATE 2017 (1 Mark): Product of eigenvalues of matrix P is

A) -6

B) 2

C) 6

D) -2

​Solution: Product of eigenvalues = |P| = 2(3 - 6) - 0 + 1(8-0) = 2

Option B is the correct answer.

Example 4:

GATE 2017 (2 Marks): Consider the matrix A whose eigenvectors corresponding to eigenvalues λ1 and λ2 are x1 and x2 respectively. The value of (transpose of x1)*(x2) is _________.

GATE 2017 (2 Marks): Consider the matrix A whose eigenvectors corresponding to eigenvalues λ1 and λ2 are x1 and x2 respectively. The value of (transpose of x1)*(x2) is _________.

Solution:

GATE 2017 (2 Marks): Consider the matrix A whose eigenvectors corresponding to eigenvalues λ1 and λ2 are x1 and x2 respectively. The value of (transpose of x1)*(x2) is _________.

= 70(λ2 - 80) + 70(λ1-50)

= 70(λ1 + λ2 - 130)

= 0

Since λ1 + λ2 = sum of diagonal elements = 50 + 80 = 130

Example 5:

GATE 2016 (2 Marks): The number of linearly independent eigenvectors of matrix A is ________.

GATE 2016 (2 Marks): The number of linearly independent eigenvectors of matrix A is ________.

Solution: Two eigenvectors of matrix A are x axis and z axis with corresponding eigenvalue as 2 and 3 respectively.

Sum of eigenvalues = Trace of matrix = 2 + 2 + 3 = λ1 + λ2 + λ3

Here λ1 = 2, λ2 = 3; Hence λ3 = 2

Since the matrix has only two distinct eigenvalues, and x-axis and z-axis are only two corresponding eigenvectors; matrix is having two linearly independent eigenvectors.

Example 6:

GATE 2016 (1 Marks): The condition for which eigenvalues of the matrix A are positive, is

GATE 2016 (1 Marks): The condition for which eigenvalues of the matrix A are positive, is

A) k>1/2

B) k>-2

C) k>0

D) k<-1/2

Solution: Suppose λ1 and λ2 are eigenvalues of matrix A.

λ1λ2 = |A| = 2k - 1>0

λ1 + λ2 = 2+k >0

k>1/2 and k>-2

Both the conditions will be true for k>1/2

Hence option A is the correct answer.

Example 7:

 

GATE 2015 (1 Mark): At least one eigenvalue of a singular matrix is

A) Positive

B) Zero

C) Negative

D) Imaginary

Solution: For a singular matrix |A| = 0, hence product of matrix is 0. This is possible only if at least one eigenvalue is zero.

Option B is the correct answer.

Example 8:

 

GATE 2014 (1 Mark):Consider a 3 × 3 real symmetric matrix S such that two of its eigen values are a ≠ 0 , b ≠ 0 with respective eigenvectors X and Y. If a ≠ b then x1y1 + x2y2 + x3y3 equals

GATE 2014 (1 Mark):Consider a 3 × 3 real symmetric matrix S such that two of its eigen values are a ≠ 0 , b ≠ 0 with respective eigenvectors X and Y. If a ≠ b then x1y1 + x2y2 + x3y3 equals

A) a

B) b

C) ab

D) 0

Solution: For symmetric matrix having two distinct eigenvalues, corresponding eigenvectors are orthogonal.

x1y1 + x2y2 + x3y3 = 0

Hence option D is the correct answer.

Example 9:

GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as

GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as
GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as
GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as
GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as
GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as

Solution: Characteristic equation of matrix is given as

|A - λI| = 0

(5 - λ)(3 - λ) - 3 = 0

λ^2 - 8λ + 12 = 0

λ = 6,2

For λ = 2

(A - 2I)X = 0

GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as

x + y = 0

y = -x

x/1 = y/-1 = k

x = k, y = -k

Eigenvector corresponding to λ = 2 is

GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as

Eigenvector in normalized form is

GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as

Option B is the correct answer.

Example 10:

GATE 2008 (2 Marks): The eigenvectors of the matrix A as given below are written in the form

GATE 2008 (2 Marks): The eigenvectors of the matrix A as given below are written in the form
GATE 2008 (2 Marks): The eigenvectors of the matrix A as given below are written in the form

What is value of a+b.

A) 0

B) 1/2

C) 1

D) 2

Solution: One eigenvalue of matrix is 1 with corresponding eigenvector as y = 0.

Sum of eigenvalue = Trace of matrix = 1+2

Hence second eigenvalue is 2. Corresponding eigenvector is find by equation

(A - 2I)X = 0

GATE 2008 (2 Marks): The eigenvectors of the matrix A as given below are written in the form

-x + 2y = 0

y = x/2

x/1 = y/(1/2)

So two eigenvectors may be written as

GATE 2008 (2 Marks): The eigenvectors of the matrix A as given below are written in the form

Hence value of a+b = 1/2

Option B is the correct answer.

Example 11:

GATE 2006 (2 Marks): Eigenvalue of a matrix S are 5 & 1. What are the eigenvalues of matrix S*S.

GATE 2006 (2 Marks): Eigenvalue of a matrix S are 5 & 1. What are the eigenvalues of matrix S*S.

A) 1 and 25

B) 6 and 4

C) 5 and 1

D) 2 and 10

Solution: If λ is eigenvalue of matrix S, eigenvalue of S*S will be λ^2 which is 25 and 1.

Option A is the correct answer.

Example 12:

GATE 2005 (2 Marks): Which one of the following is an eigenvector of the matrix.

GATE 2005 (2 Marks): Which one of the following is an eigenvector of the matrix.
GATE 2005 (2 Marks): Which one of the following is an eigenvector of the matrix.
GATE 2005 (2 Marks): Which one of the following is an eigenvector of the matrix.
GATE 2005 (2 Marks): Which one of the following is an eigenvector of the matrix.
GATE 2005 (2 Marks): Which one of the following is an eigenvector of the matrix.

Solution: As discussed above, eigenvector when transformed remains on its span.

For option A

GATE 2005 (2 Marks): Which one of the following is an eigenvector of the matrix.

This vector remains on its line and is only enlarged by a factor 5.

Hence option A is the correct answer.

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