{"id":74,"date":"2011-10-26T06:51:31","date_gmt":"2011-10-26T06:51:31","guid":{"rendered":"http:\/\/florian-roemer.de\/blog\/?p=74"},"modified":"2012-01-22T16:22:45","modified_gmt":"2012-01-22T16:22:45","slug":"and-a-simpler-proof-for-it","status":"publish","type":"post","link":"https:\/\/florian-roemer.de\/blog\/and-a-simpler-proof-for-it\/","title":{"rendered":"&#8230; and a simpler proof for it"},"content":{"rendered":"<p>A much simpler proof for the <a title=\"Even more algebra fun\" href=\"http:\/\/florian-roemer.de\/blog\/?p=70\">more generic algebraic rule on manipulating quadratic forms<\/a> I posted recently just became apparent to me. All you need to do is to first show that<\/p>\n<p>$${\\rm trace}\\{\\ma{A}^{\\rm H} \\cdot \\ma{B}\\} = {\\rm vec}\\{\\ma{A}\\}^{\\rm H} \\cdot {\\rm vec}\\{\\ma{B}\\},$$<\/p>\n<p>which is fairly easy because both represent a short-hand notation for $\\sum_k\\sum_\\ell\\sum_m a_{\\ell,k}^* b_{\\ell,m}$. Once you have this rule set you simply break ${\\rm trace}\\left\\{\\ma{A} \\cdot \\ma{X} \\cdot \\ma{R} \\cdot \\ma{X}^{\\rm H} \\cdot \\ma{B}^{\\rm H}\\right\\}$ into ${\\rm vec}\\left\\{\\left(\\ma{A} \\cdot \\ma{X}\\right)^{\\rm H}\\right\\}^{\\rm H} \\cdot {\\rm vec}\\left\\{\\ma{R} \\cdot \\ma{X}^{\\rm H} \\cdot \\ma{B}^{\\rm H}\\right\\}$ and apply the <a title=\"Fun with algebra\" href=\"http:\/\/florian-roemer.de\/blog\/?p=41\">rules for generic linear forms twice<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A much simpler proof for the more generic algebraic rule on manipulating quadratic forms I posted recently just became apparent to me. All you need to do is to first show that $${\\rm trace}\\{\\ma{A}^{\\rm H} \\cdot \\ma{B}\\} = {\\rm vec}\\{\\ma{A}\\}^{\\rm H} \\cdot {\\rm vec}\\{\\ma{B}\\},$$ which is fairly easy because both represent a short-hand notation for [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3],"tags":[4,5,7],"_links":{"self":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/74"}],"collection":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/comments?post=74"}],"version-history":[{"count":5,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/74\/revisions"}],"predecessor-version":[{"id":77,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/74\/revisions\/77"}],"wp:attachment":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/media?parent=74"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/categories?post=74"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/tags?post=74"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}