{"id":13,"date":"2023-05-12T14:22:43","date_gmt":"2023-05-12T13:22:43","guid":{"rendered":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/?p=13"},"modified":"2023-07-21T06:18:56","modified_gmt":"2023-07-21T05:18:56","slug":"wronskian","status":"publish","type":"post","link":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/2023\/05\/12\/wronskian\/","title":{"rendered":"Wronskian"},"content":{"rendered":"<h3><em>1. Derivative of a determinant<\/em><\/h3>\n<p>Consider the determinant <em>W<sub>(t)<\/sub> <\/em> of a <em>n\u00d7n<\/em> matrix <em>Y<\/em> which each element is a function of <em>t<\/em>. Assume elements to be independent variables. Then we could write:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"46\" class=\"alignnone wp-image-84\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1.png\" alt=\"\" \/><\/p>\n<p>Where <em>C<sub>ij<\/sub><\/em>\u00a0are the corresponding cofactors. Thus we have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1368\" height=\"62\" class=\"alignnone size-full wp-image-85\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation2.png\" alt=\"\" \/><\/p>\n<p>Let&#8217;s define \u03b3<em><sub><em>i<\/em><\/sub><\/em> as the new matrices form by substituting the ith row with its derivative. Then we could write (2) in a tidier form:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1368\" height=\"56\" class=\"alignnone size-full wp-image-86\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation3.png\" alt=\"\" \/><\/p>\n<h3><em>2. Abel-Jacobi-Liouville identity <\/em><\/h3>\n<p>As we know, any system of linear ordinary equations can be extracted in to a single linear equation, namely:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1368\" height=\"56\" class=\"alignnone size-full wp-image-87\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation4.png\" alt=\"\" \/><\/p>\n<p>And it is followed by that,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"56\" class=\"alignnone size-full wp-image-88\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation5.png\" alt=\"\" \/><\/p>\n<p>So we obseverve that a particular <strong>row<\/strong> of the derivative is the linear combination of the original rows, since for the kth row of <em>Y<\/em>, different elements on jth column are multiplied by the same factor <em>A<\/em><sub><em>ik<\/em><\/sub>. So of course, each term on the right hand side of (3) will be <em>W<\/em> times the corresponding element of <em>A<sub>ii<\/sub><\/em>.<br \/>\nThus,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1368\" height=\"56\" class=\"alignnone size-full wp-image-89\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation6.png\" alt=\"\" \/><\/p>\n<p>This has resulted in some interesting conclusions. For example if the solutions are independent for any point within the domin, they must be independent entirely.<\/p>\n<p>[Literature: Pontryagain 1962, Chapter 3]<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1. Derivative of a determinant Consider the determinant W(t) of a n\u00d7n matrix Y which each element is a function of t. Assume elements to be independent variables. Then we could write: Where Cij\u00a0are the corresponding cofactors. Thus we have: Let&#8217;s define \u03b3i as the new matrices form by substituting the ith row with its [&hellip;]<\/p>\n","protected":false},"author":1741,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[19,2],"tags":[6,5,4],"class_list":["post-13","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","category-ordinary-differential-equations","tag-abel-jacobi-liouville-identity","tag-derivatives-of-determinants","tag-wronskian"],"_links":{"self":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/13","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/users\/1741"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/comments?post=13"}],"version-history":[{"count":18,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/13\/revisions"}],"predecessor-version":[{"id":100,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/13\/revisions\/100"}],"wp:attachment":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/media?parent=13"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/categories?post=13"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/tags?post=13"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}