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.
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!
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!