# Symmetric Matrix - Eigenvectors, Eigenvalues

• March 24th 2012, 08:40 AM
shinn
Symmetric Matrix - Eigenvectors, Eigenvalues
Hi, just wondering if anyone would be able to assist me with this question.

Assume that http://chart.apis.google.com.nyud.ne...5Ctimes%20n%7D is symmetric. Let http://chart.apis.google.com.nyud.ne...ts%20%2C%20v_n be an orthonormal basis of eigenvectors with corresponding (real) eigenvalues http://chart.apis.google.com.nyud.ne...%2C%5Clambda_n.

Show that the 2-norm of any vector http://chart.apis.google.com.nyud.ne...0%5Cin%20R%5En satisfies

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

Any help is appreciated!
• March 24th 2012, 09:35 AM
thm43608
Re: Symmetric Matrix - Eigenvectors, Eigenvalues
This isn't my area, but since $A$ is symmetric, isn't $v_j=v_j^T$? That would get you the $\left( v_j^T \right)^2$ part. I'm not sure if this helps any or not.
• March 24th 2012, 05:28 PM
shinn
Re: Symmetric Matrix - Eigenvectors, Eigenvalues
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 http://chart.apis.google.com.nyud.ne...q%20%5Cdots%20.

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!