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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).
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Such vectors are called eigenvectors and the corresponding value through which magnitude changes is called eigenvalue.
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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
Vector 3j remains on y axis even after transformation and only its magnitude changes by 2. Similarly all vectors on yaxis retain its orientation and are only enlarged by a value 2. Hence, x = 0 is eigenvector with corresponding eigenvalue as 2.
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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 nonzero 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).
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Notes:

A square matrix A and its transpose have same eigenvalues.

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

Product of eigenvalues is equal to determinant of matrix.

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

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

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

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

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:
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.
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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
Solution: Characteristics equation of the matrix P is
P  λI = 0
(1λ)[(1λ)(1λ)  0] = 0
λ = 1,1,1
So the equation has only one distinct eigenvalue.
Option B is the correct answer.
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Example 2:
GATE 2019 (1 Mark): In matrix equation [A]{X} = {R}, one of the eigenvalues of matrix A is
A) 4
B) 8
C) 15
D) 16
â€‹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.
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Example 3:
GATE 2017 (1 Mark): Product of eigenvalues of matrix P is
A) 6
B) 2
C) 6
D) 2
â€‹Solution: Product of eigenvalues = P = 2(3  6)  0 + 1(80) = 2
Option B is the correct answer.
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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 _________.
Solution:
= 70(λ2  80) + 70(λ150)
= 70(λ1 + λ2  130)
= 0
Since λ1 + λ2 = sum of diagonal elements = 50 + 80 = 130
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Example 5:
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 xaxis and zaxis are only two corresponding eigenvectors; matrix is having two linearly independent eigenvectors.
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Example 6:
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
â€‹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.
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Example 7:
GATE 2015 (1 Mark): At least one eigenvalue of a singular matrix is
A) Positive
B) Zero
C) Negative
D) Imaginary
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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
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.
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Example 9:
GATE 2012 (2 Marks): For the matrix A, one of the normalized eigenvector is given as
â€‹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
x + y = 0
y = x
x/1 = y/1 = k
x = k, y = k
Eigenvector corresponding to λ = 2 is
Eigenvector in normalized form is
Option B is the correct answer.
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Example 10:
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
x + 2y = 0
y = x/2
x/1 = y/(1/2)
So two eigenvectors may be written as
Hence value of a+b = 1/2
Option B is the correct answer.
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Example 11:
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.
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Example 12:
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
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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