{"id":79291,"date":"2021-12-03T02:26:37","date_gmt":"2021-12-03T02:26:37","guid":{"rendered":"https:\/\/papersspot.com\/blog\/2021\/12\/03\/solution-since-a-is-idempotent-we-have-a%c2%b2-a-i\/"},"modified":"2021-12-03T02:26:37","modified_gmt":"2021-12-03T02:26:37","slug":"solution-since-a-is-idempotent-we-have-a%c2%b2-a-i","status":"publish","type":"post","link":"https:\/\/papersspot.com\/blog\/2021\/12\/03\/solution-since-a-is-idempotent-we-have-a%c2%b2-a-i\/","title":{"rendered":"Solution: Since A is idempotent, we have A\u00b2 = A (I \u2013"},"content":{"rendered":"<p>Solution:<\/p>\n<p> Since A is idempotent, we have <\/p>\n<p> A\u00b2 = A<\/p>\n<p> (I \u2013 A) \u00b2 = I\u00b2 + A\u00b2 &#8211; 2IA<\/p>\n<p> (I \u2013 A) \u00b2 = I + A &#8211; 2A<\/p>\n<p> (I \u2013 A) \u00b2 = I -A <\/p>\n<p> \u300b I \u2013 A is idempotent<\/p>\n<p> Since A is inevitable, A\u2010\u00b9 exists<\/p>\n<p> A\u00b2 = A<\/p>\n<p> Multiply both sides by A-\u00b9<\/p>\n<p> A-\u00b9(AA) = A-\u00b9A<\/p>\n<p> (A-\u00b9A) A = A-\u00b9A<\/p>\n<p> IA = I<\/p>\n<p> \u300b A= I<\/p>\n<p> Let \u03bb be the given value of A corresponding to the vector.<\/p>\n<p> Then, Av = \u03bbv<\/p>\n<p> \u300b A\u00b3v = \u03bbv <\/p>\n<p> \u300b A(Av) = \u03bbv<\/p>\n<p> \u300b A(\u03bbv) = \u03bbv<\/p>\n<p> \u300b\u03bb (Av) = \u03bbv<\/p>\n<p> \u300b\u03bb (\u03bbv) = \u03bbv <\/p>\n<p> \u300b\u03bb\u00b2v = \u03bbv<\/p>\n<p> \u300b\u03bb\u00b2v \u2013 \u03bbv = 0<\/p>\n<p> \u300b (\u03bb\u00b2-\u03bb) v = 0<\/p>\n<p> But v \u2260 0<\/p>\n<p> \u300b\u03bb\u00b2 &#8211; \u03bb = 0<\/p>\n<p> \u300b\u03bb (\u03bb-1) = 0<\/p>\n<p> \u300b\u03bb = 0, 1<\/p>\n<p> \u10fb The only eigenvalues of A are 0 and 1<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution: Since A is idempotent, we have A\u00b2 = A (I \u2013 A) \u00b2 = I\u00b2 + A\u00b2 &#8211; 2IA (I \u2013 A) \u00b2 = I + A &#8211; 2A (I \u2013 A) \u00b2 = I -A \u300b I \u2013 A is idempotent Since A is inevitable, A\u2010\u00b9 exists A\u00b2 = A Multiply both sides [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[10],"class_list":["post-79291","post","type-post","status-publish","format-standard","hentry","category-research-paper-writing","tag-writing"],"_links":{"self":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/posts\/79291","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/comments?post=79291"}],"version-history":[{"count":0,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/posts\/79291\/revisions"}],"wp:attachment":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/media?parent=79291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/categories?post=79291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/tags?post=79291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}