Thread: Symmetric Matrix - Eigenvectors, Eigenvalues

Hi, just wondering if anyone would be able to assist me with this question.

Assume that is symmetric. Let be an orthonormal basis of eigenvectors with corresponding (real) eigenvalues .

Show that the 2-norm of any vector satisfies
.

Note, I have shown in part a of the question that:


Any help is appreciated!

This isn't my area, but since is symmetric, isn't ? That would get you the part. I'm not sure if this helps any or not.

Thank you for your attempt, but I figured it out.

Setting V as the matrix with columns v_j you can write this in matrix notation. v_j are orthonormal, or in matrix notation V^t V = I which also means VVt = I. |x|^2 = x^t x, and the j'th entry of V^t x is v_j^t x, and:

|x|^2 = x^t x = x^t I x = x^t V V^t x = (V^t x)^t (V^t x)=|V^t x|^2.

But, I also have a follow up question to this which is to show that [ ||Ax || \leq |\lambda_1| ||x|| ] where the eigenvalues are ordered such that .

I'm guessing we need to use the previous part of the question, but after substituting Ax into the LHS, I get stuck showing the inequality.

Would it possible to give me some idea on this one? Thanks!