# Thread:

1. ## 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!

2. ## 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.

3. ## 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 .

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!
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9180467128753662, "perplexity_flag": "middle"}