I am given the following problem: A = [ <mtable rowspacing="4pt" columnspaci

Sattelhofsk 2022-06-26 Answered
I am given the following problem:
A = [ 4 1 1 1 2 3 1 3 2 ]
Find
max x | ( A x , x ) | ( x , x )
where ( . , . ) is a dot product of vectors and the maximization is performed over all x = [ x 1 x 2 x 3 ] T R 3 , such that i = 1 3 x i = 0
I have found the eigenvectors for A and they happen to match the sum criterion:
E ( λ 1 ) = span ( [ 2 1 1 ] T )
for λ 1 = 3 and
E ( λ 2 ) = span ( [ 0 1 1 ] T )
for λ 2 = 1.
(For λ 3 = 6 there are no eigenvectors).
Can the above eigenvectors and eigenvalues be used for solving this maximization problem?
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Answers (1)

Xzavier Shelton
Answered 2022-06-27 Author has 26 answers
You wrote "for λ 3 = 6 there are no eigenvectors". This is not true !
We have, since A is symmetric, that max x | ( A x , x ) | ( x , x ) = max { λ : λ σ ( A ) }, where σ ( A ) denotes the set of eigenvalues of A.
Hence
max x | ( A x , x ) | ( x , x ) = 6.

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