GATE_2014 ME

Question

Consider a 3 x 3 real symmetric matrix S such that two of its eigen values are  a ≠ 0 ,  b ≠ 0 with respective Eigen vectors 

[ x₁ x₂ x₃ ]  , [ y₁ y₂ y₃ ] . 

If a ≠ b then x₁y₁   + x₂y₂ + x₃y₃  is

a)   a

b)   b

c)    ab

d)    0

Answer 1

Let us assume two of eigen values of S are  a ≠ 0 ,  b ≠ 0 with X and Y eigen vectors respectively.

Y^TX = x₁y₁   + x₂y₂ + x₃y₃

Since X is an eigen vector, therefore, 

SX = aX

Y^TSX = aY^TX   (premultiplying Y^T on both sides)

(SY)^TX = aY^TX ( S is a real symmetric matrix)

bY^TX = aY^TX

(b-a)Y^TX = 0

Y^TX = 0   ( given that a != b)

 Eigen vectors X and Y are orthogonal to each other.

Therefore, Correct answer would be (d).

Comments

What is the purpose of taking Y^TX in the first Line... I mean why to take this in these cases?

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Just to show that we want to calculate Y^TX.

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How could you write like this ?
Are you trying to summ up the Eigen values of Y^TX
Y^TX = x₁y₁   + x₂y₂ + x₃y₃

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awesome explanation