In this problem, represents the set of real numbers, and represents the set of real column vectors of length . For , denotes its transpose. Let be the identity matrix.
Consider an eigensystem of a real symmetric matrix ,
where and are an eigenvalue and a corresponding eigenvector, respectively.
Let be the maximum of eigenvalues of matrix .
You may use the following facts on the eigenvalues and the eigenvectors of a real symmetric matrix without proofs:
There are independent eigenvectors that form an orthogonal basis.
Every eigenvalue is a real number.
Answer the following questions.
(1) Prove that if is an eigenvector of , it is also an eigenvector of for any .
(2) Prove that
(3) Prove that
for any .
(4) Suppose that matrix is also an real symmetric matrix. Prove that
Let be a unit vector which maximizes .
Because is real and symmetric, then it can be diagonalized , where is orthogonal matrix, which columns form orthogonal basis.
In following equation, let's substitute :
On the other hand, let be eigenvector corresponding to .
Because maximizes , then:
Let .
From question (2) we know that for arbitrary : look at second to the last line of Q2 calculations.
Now if we know that fact, we get the inequality trivially: