{"id":1564,"date":"2024-03-29T19:04:06","date_gmt":"2024-03-29T19:04:06","guid":{"rendered":"https:\/\/sinatootoonian.com\/?p=1564"},"modified":"2025-12-27T16:05:31","modified_gmt":"2025-12-27T16:05:31","slug":"wrangling-quartics-v","status":"publish","type":"post","link":"https:\/\/sinatootoonian.com\/index.php\/2024\/03\/29\/wrangling-quartics-v\/","title":{"rendered":"Wrangling quartics, V"},"content":{"rendered":"\n

Yesterday I went to discuss the problem with one of my colleagues. He had the interesting idea of modelling $S$, and especially $S^2$, as low rank, in particular as $S = s_1 e_1 e_1^T$. That is, shifting the focus on $S$ from $Z$. I tried this out today, and although it didn’t quite pan out, the discussion we had sent me on a useful trajectory that I will describe in this post.<\/p>\n\n\n\n

As always in this series, the equation whose solutions we’re trying to understand is \\begin{align}S^2 Z^2 S^2 – S C S = \\lambda (Z^{-1} – I).\\label{main}\\tag{1}\\end{align} I returned to approximating $Z$, and considered approximations for which $Z^2$ and $Z^{-1}$ are in the same ‘family’. For example, if $Z \\approx I + uu^T$, then
\\begin{align} Z^2 &= I + (2 + u^T u) uu^T, \\\\Z^{-1} &= I – {1 \\over 1 + u^T u} uu^T.\\end{align} Both of these are of the form $I + \\beta uu^T$, so there might be some hope of equating coefficients of $uu^T$ on the left and right sides of $\\Eqn{main}$. When I plugged in this form for $Z$, I was able to derive an interesting eigenvalue relationship that $u$ satisfies.<\/p>\n\n\n\n

I found earlier in the week that $Z$ can be approximated as a low-rank update to a diagonal matrix, as shown on the far left below:<\/p>\n\n\n\n

\"\"
Approximation of $Z$ (left) by a multiple of the identity, a diagonal matrix, and rank 1 updates to these.<\/em><\/figcaption><\/figure>\n\n\n\n

Unfortunately, the diagonal matrix is not quite the identity. In particular, its first value is very close to zero:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

However, the success I had with deriving an eigenvalue equation for the $I + uu^T$ approximation motivated me to try to derive one here as well. <\/p>\n\n\n\n

We first need $Z^2$ and $Z^{-1}$:
\\begin{align} Z &\\approx D + uu^T, \\\\
Z^2 &\\approx D^2 +D uu^T + uu^T D + (u^T u) uu^T,\\\\
Z^{-1} &\\approx D^{-1} – {D^{-1} uu^T D^{-1} \\over 1 + u^T D u}.\\end{align}<\/p>\n\n\n\n

Plugging these into $\\Eqn{main}$ gives
$$ {S^2 (D^2 + D uu^T + uu^T D + (u^T u) uu^T )S^2 – S C S \\over \\lambda} + I = D^{-1} – {D^{-1} u u^T D^{-1} \\over 1 + u^T D^{-1} u}.$$<\/p>\n\n\n\n

Multiplying both sides by $u$ gives
$$ {(\\Lambda_1 + \\Lambda_2 + \\Lambda_3 + \\Lambda_4) u – S C S u \\over \\lambda} = \\left[\\left(1 – {u^T D^{-1} u\\over 1 + u^T D^{-1} u}\\right) D^{-1} – I\\right]u,$$ where I’ve defined the diagonal matrices \\begin{align} \\Lambda_1 &= S^2 D^2 S^2, & \\Lambda_2 &= (u^T S^2 u)S^2 D,\\\\ \\Lambda_3 &= (u^T D S^2 u) S^2, & \\Lambda_4 &= (u^T u) (u^T S^2 u) S^2.\\end{align}<\/p>\n\n\n\n

Then, largely because of how small $D_{11}$ is, a few amazing things happen. <\/p>\n\n\n\n

First, the matrix on righthand side becomes approximately $-I + 2e_1e_1^T$. <\/p>\n\n\n\n

Second, the elements of $\\Lambda_4$, especially the first few, are much larger than the rest, so we can approximate $$ \\sum_{i=1}^4 \\Lambda_i \\approx \\Lambda_4.$$ Although e.g. $\\Lambda_1$ has $S^4$ in it, the presence of $D^2$ appears enough to squash its influence. $\\Lambda_4$ is the only term that doesn’t have $D$ in it, and it dominates the others.<\/p>\n\n\n\n

Third, $|\\Lambda_4 \/ \\lambda| \\gg 1$ for almost all the elements, so we can ignore the righthand side. <\/p>\n\n\n\n

So, after these three amazing effects, we end up with
$$ \\Lambda_4 u = (u^T u) (u^T S^2 u) S^2 u \\approx S C S u.$$ Left multiplying by $S^{-1}$ we get $$ (u^T u) (u^T S^2 u) S u \\approx C S u.$$ In other words, $S u$ is an eigenvector of $C$ with eigenvalue $(u^T u) (u^T S^2 u).$<\/p>\n\n\n\n

Not quite sure what this all means, but an interesting result. <\/p>\n\n\n\n

$$\\begin{flalign*} && \\phantom{a} & \\hfill \\square \\end{flalign*}$$
<\/p>\n","protected":false},"excerpt":{"rendered":"

Yesterday I went to discuss the problem with one of my colleagues. He had the interesting idea of modelling $S$, and especially $S^2$, as low rank, in particular as $S = s_1 e_1 e_1^T$. That is, shifting the focus on $S$ from $Z$. I tried this out today, and although it didn’t quite pan out, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1,148],"tags":[48,49,40],"class_list":["post-1564","post","type-post","status-publish","format-standard","hentry","category-blog","category-research","tag-eigenvalues","tag-low-rank","tag-work"],"acf":[],"_links":{"self":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts\/1564","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/comments?post=1564"}],"version-history":[{"count":26,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts\/1564\/revisions"}],"predecessor-version":[{"id":1592,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts\/1564\/revisions\/1592"}],"wp:attachment":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/media?parent=1564"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/categories?post=1564"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/tags?post=1564"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}