Symmetric Matrix - Eigenvectors, Eigenvalues

Re: Symmetric Matrix - Eigenvectors, Eigenvalues

This isn't my area, but since $\displaystyle A$ is symmetric, isn't $\displaystyle v_j=v_j^T$? That would get you the $\displaystyle \left( v_j^T \right)^2$ part. I'm not sure if this helps any or not.

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!