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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
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Eigen Vector Equation = AX = λX = A − λI X = 0 , where X is the Eigen vector.

X X ′
Here, there are two Eigen vectors: 1 and 1
X 2 X ′ 2
Let us now find the first Eigen vector. Expanding the Eigen vector equation, we have:

50 − λ 10 X 0
= 1 1 = - - - - (1.71)
10 25 − λ 1 X 2 0

where X and X are components of the first Eigen vector.
1
2
Here, the values of X and X can not be determined, as, the equations are homogenous, that is,
1
2
RHS of the equations are zero. Only the relationship between X and X can be got by solving
1
2
either equations (1.72) or (1.73) that is got by simplifying equation (1.71).
50 − λ X + 10 ∗ X = 0 - - - - (1.72)
1
2
1
(10 ∗ X ) + (25 − λ )X = 0 - - - - (1.73)
1
1
2
Let us simplify equation (1.72) to compute a relationship between X and X .
1
2
10 ∗ X 10 ∗ X
2
2
X = − = X = − = X = 2.849 X
1
1
1
2
50 − λ 50 − 53.51
1
2.849
Substituting (X = 1), we get the components of the first Eigen vector to be: 1
2
The second Eigen vector is got by replacing (λ ) with (λ ) and the first Eigen vector with the second
1
2
in equations (1.71).
50 − λ 10 X ′ 0
= 2 1 = - - - - (1.74)
10 25 − λ 2 X ′ 2 0
Simplifying equation (1.74), we get:
′
′
50 − λ X + 10 ∗ X = 0 - - - - (1.75)
2
1
2
′
′
(10 ∗ X ) + (25 − λ )X = 0 - - - - (1.76)
1
2
2

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 57