GATE | Gate IT 2007 | Question 2

Let A be the \begin{pmatrix} 3&1&1&2\\ 3&1&1&2 \end{pmatrix} . What is the maximum value of xTAx where the maximum is taken over all x that are the unit eigenvectors of A?
(A) 5
(B) (5 + √5)/2
(C) 3
(D) (5 – √5)/2


Answer: (B)

Explanation: |M-λ.I| = 0, where λ is the eigen values and I is the identity matrix
|A-(λ*I)| = 0
(3-λ)(2-λ)-1 = 0
6-3λ -2λ + λ2+1=0
λ2-5λ+5=0
λ = (5+√5)/2 and (5-√5)/2,
λ = (5+√5)/2 is max value another root with negative sign which will not be max value.
For, λ=5+5√2, xTAx=[18.131 21.231 21.231 34.331]
For, λ=5−5√2, xTAx=\begin{pmatrix} 14.300&−0.700\\ 6.200&1.200 \end{pmatrix}
Hence, for λ=5+5√2 the value of xTAx is maximum.

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