{"id":160,"date":"2007-05-18T15:14:22","date_gmt":"2007-05-18T15:14:22","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=160"},"modified":"2020-03-14T19:29:30","modified_gmt":"2020-03-14T19:29:30","slug":"eigenvalues-without-determinants","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=160","title":{"rendered":"Eigenvalues without determinants"},"content":{"rendered":"

Most (all?) undergraduate courses use determinants to introduce eigenvalues and eigenvectors. So the eigenvalues of a matrix A (or linear transformation t) are the solutions of \\det(A-\\lambda I)=0. However, Sheldon Axler published a paper in 1994 called Down with Determinants!<\/a> where he maintains that determinants should not be used so early in linear algebra courses. He gives a very nice proof of the existence of eigenvalues in finite-dimensional vector spaces (over \\mathbb{C}) which I would like to reproduce here.<\/p>\n

Every linear transformation of a finite-dimensional complex vector space has an eigenvalue.<\/em><\/p><\/blockquote>\n

Here is his proof:<\/p>\n

Let V be a non-trivial finite-dimensional complex vector space and t<\/em>\u00c2\u00a0a linear transformation t\\colon V \\to V. Let v be a fixed non-zero vector in V and suppose that \\dim V =n. Then the n+1 vectors v,t(v),t^2 (v),\\dots,t^n (v) are linearly dependent. Hence there exists complex numbers \\alpha_0,\\alpha_1,\\dots,\\alpha_n not all 0 such that<\/p>\n

\\alpha_0 v + \\alpha_1 t(v) + \\dots + \\alpha_n t^n (v)=0.<\/p>\n

and hence<\/p>\n

(\\alpha_01+ \\alpha_1 t + \\dots + \\alpha_n t^n)(v)=0.<\/p>\n

Now, since \\mathbb{C} is algebraically closed, \u00c2\u00a0the polynomial \\alpha_0 + \\alpha_1 z+ \\dots + \\alpha_n z^n will factorise so we get<\/p>\n

\\alpha_0 + \\alpha_1 z+ \\dots + \\alpha_n z^n=c(z-\\lambda_1)(z-\\lambda_2)\\dots(z-\\lambda_m).<\/p>\n

where c, \\lambda_1,\\dots,\\lambda_m are complex numbers with c \\neq 0. It follows that<\/p>\n

c(t-\\lambda_1 1)(t-\\lambda_2 1)\\dots(t-\\lambda _m 1)(v)=0<\/p>\n

which means that, since this is composition of functions, and v \\ne 0, then
\neither<\/em>
\n(t-\\lambda _m 1)(v)=0 so\u00c2\u00a0\u00c2\u00a0(t-\\lambda_m 1)(v)=v\u00c2\u00a0so that t(v)=\\lambda_m v and v is an eigenvector
\nor<\/em>
\n(t-\\lambda _{m-1} 1)(t-\\lambda _m 1)(v)=0 so (t-\\lambda _m 1)(v)\u00c2\u00a0is an eigenvector
\nor
\n…
\nor
\n<\/em>(t-\\lambda_2 1)\\dots(t-\\lambda _m 1)(v) is an eigenvector,
\nand hence t has an eigenvalue. \u00c2\u00a0\u00c2\u00a0\\blacksquare<\/p>\n

Discussion on this approach of not using determinants can be found at NeverEndingBooks<\/a>\u00c2\u00a0and The n-category Caf\u00c3\u00a9<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Most (all?) undergraduate courses use determinants to introduce eigenvalues and eigenvectors. So the eigenvalues of a matrix (or linear transformation ) are the solutions of . However, Sheldon Axler published a paper in 1994 called Down with Determinants! where he maintains that determinants should not be used so early in linear algebra courses. He gives […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-160","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=160"}],"version-history":[{"count":1,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/160\/revisions"}],"predecessor-version":[{"id":286,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/160\/revisions\/286"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}