{"id":5314,"date":"2014-04-28T06:05:35","date_gmt":"2014-04-27T21:05:35","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5314"},"modified":"2014-08-01T18:25:49","modified_gmt":"2014-08-01T09:25:49","slug":"operator-interpretation-of-matrices","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5314","title":{"rendered":"Operator interpretation of matrices"},"content":{"rendered":"<div class=\"theContentWrap-ccc\"><blockquote>\n<p>If <img src='https:\/\/s0.wp.com\/latex.php?latex=A&#038;bg=T&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' \/> is an <img src='https:\/\/s0.wp.com\/latex.php?latex=n+%5Ctimes+n&#038;bg=T&#038;fg=000000&#038;s=0' alt='n \\times n' title='n \\times n' class='latex' \/> matrix, we can think of it as an <em>operator<\/em> or <em>transformation<\/em> acting on a column vector <img src='https:\/\/s0.wp.com\/latex.php?latex=X&#038;bg=T&#038;fg=000000&#038;s=0' alt='X' title='X' class='latex' \/> to produce <img src='https:\/\/s0.wp.com\/latex.php?latex=AX&#038;bg=T&#038;fg=000000&#038;s=0' alt='AX' title='AX' class='latex' \/> which is another column vector. With this interpretation equation (21) asks for those vectors <img src='https:\/\/s0.wp.com\/latex.php?latex=X&#038;bg=T&#038;fg=000000&#038;s=0' alt='X' title='X' class='latex' \/> which are transformed by <img src='https:\/\/s0.wp.com\/latex.php?latex=A&#038;bg=T&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' \/> into constant multiples of themselves [or equivalently into vectors which have the same direction but possibly different magnitude]. <\/p>\n<p>If case <img src='https:\/\/s0.wp.com\/latex.php?latex=A&#038;bg=T&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' \/> is an orthogonal matrix, the transformation is a <em>rotation<\/em> and explains why the absolute value of all the eigenvalues in such case are equal to one, since an ordinary rotation of a vector would not change its magnitude. <\/p>\n<p>The ideas of transformation are very convenient in giving interpretations to many properties of matrices. <\/p>\n<\/blockquote>\n<p><iframe src=\"\/\/rcm-fe.amazon-adsystem.com\/e\/cm?lt1=_blank&#038;bc1=000000&#038;IS2=1&#038;bg1=FFFFFF&#038;fc1=000000&#038;lc1=0000FF&#038;t=fujiitoshiki-22&#038;o=9&#038;p=8&#038;l=as4&#038;m=amazon&#038;f=ifr&#038;ref=ss_til&#038;asins=0071635408\" style=\"width:120px;height:240px;\" scrolling=\"no\" marginwidth=\"0\" marginheight=\"0\" frameborder=\"0\"><\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>If is an matrix, we can think of it as an operator or transformation acting on a column vector to produce whic &hellip; <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5314\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;Operator interpretation of matrices&#8221; \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5314","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5314","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5314"}],"version-history":[{"count":3,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5314\/revisions"}],"predecessor-version":[{"id":5470,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5314\/revisions\/5470"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5314"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5314"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}