{"id":432,"date":"2012-09-30T05:00:50","date_gmt":"2012-09-30T12:00:50","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=432"},"modified":"2012-09-29T20:02:20","modified_gmt":"2012-09-30T03:02:20","slug":"the-parity-decomposition-trick","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/09\/the-parity-decomposition-trick\/","title":{"rendered":"The Parity Decomposition Trick"},"content":{"rendered":"<p>Earlier this year, Robert Hasner showed me something that I assume everyone else (&quot;everyone else&quot;) already knows, but which <em>I<\/em> didn't know: every function on \u211d can be decomposed into the sum of an even function and an odd function&mdash;<\/p>\n<p><!--more--><\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_75b5f979d1f3514e8e2192e0cc3a1284.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"f(x) = \\frac{f(x)+f(-x)}{2} + \\frac{f(x)-f(-x)}{2}\" \/><\/span><script type='math\/tex;  mode=display'>f(x) = \\frac{f(x)+f(-x)}{2} + \\frac{f(x)-f(-x)}{2}<\/script><\/p><\/p>\n<p>(In fact, as I later read elsewhere, there's nothing essentially <em>twoful<\/em> about this idea (at least, if you don't care about restricting yourself to \u211d): you can split a function into a sum of <em>n<\/em> functions <em>f<sub>j<\/sub><\/em> for <em>j<\/em> \u2208 {0, ..., <em>n<\/em>\u20131} such that <em>f<sub>j<\/sub><\/em>(\u03c9<em>z<\/em>) = \u03c9<sup><em>j<\/em><\/sup><em>f<sub>j<\/sub><\/em>(<em>z<\/em>) where \u03c9 is an <em>n<\/em>th root of unity.)<\/p>\n<p>I started seeing the same pattern in my reading, too. Like, every matrix can be decomposed into the sum of a symmetric and a skew-symmetric matrix:<\/p>\n<p><em>A<\/em> = \u00bd(<em>A<\/em> + <em>A<\/em><sup>T<\/sup>) + \u00bd(<em>A<\/em> &ndash; <em>A<\/em><sup>T<\/sup>)<\/p>\n<p>(In fact, I have been given to understand that this observation is actually expressing a deep truth about the nature of linear transformations: every linear transformation is in some sense\u2014which I hope to make more explicit later\u2014the sum of a scaling in orthogonal directions (from the symmetric matrix; consider the spectral theorem) and a rotation (from the skew-symmetric matrix, which is said to represent an infinitesimal rotation).)<\/p>\n<p>Also (and <em>probably<\/em> related to the matrix thing), in the geometric algebra, the geometric product of vectors can be expressed as the sum of an inner product and an <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/09\/blades\/\">anticommutative outer product<\/a>.<\/p>\n<p>Are there more examples of this theme of splitting something into symmetric and antisymmetric parts? Is there a general theorem explaining exactly which mathematical objects do this kind of thing?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Earlier this year, Robert Hasner showed me something that I assume everyone else (&quot;everyone else&quot;) already knows, but which I didn't know: every function on \u211d can be decomposed into the sum of an even function and an odd function&mdash;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/432"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=432"}],"version-history":[{"count":16,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/432\/revisions"}],"predecessor-version":[{"id":448,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/432\/revisions\/448"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=432"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=432"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=432"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}