GATE | Gate IT 2007 | Question 2

Let A be the . What is the maximum value of x^TAx 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, x^TAx=[18.131 21.231 21.231 34.331]
For, λ=5−5√2, x^TAx=
Hence, for λ=5+5√2 the value of x^TAx is maximum.
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