{"id":5214,"date":"2012-10-14T13:08:36","date_gmt":"2012-10-14T11:08:36","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=5214"},"modified":"2012-10-14T13:17:29","modified_gmt":"2012-10-14T11:17:29","slug":"determinant-of-block-matrices","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2012\/10\/14\/determinant-of-block-matrices\/","title":{"rendered":"Determinant of block matrices"},"content":{"rendered":"<p style=\"text-align: center;\"><a rel=\"attachment wp-att-1822\" href=\"http:\/\/djalil.chafai.net\/blog\/2011\/05\/30\/some-nonlinear-formulas-in-linear-algebra\/matrix\/\"><img loading=\"lazy\" class=\"size-medium wp-image-1822 aligncenter\" title=\"Matrix\" src=\"\/blog\/wp-content\/uploads\/2011\/05\/matrix-300x230.jpg\" alt=\"Matrix\" width=\"300\" height=\"230\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2011\/05\/matrix-300x230.jpg 300w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2011\/05\/matrix.jpg 463w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p style=\"text-align: justify;\">This is a tiny followup of a <a href= \"\/blog\/2011\/05\/30\/some-nonlinear-formulas-in-linear-algebra\/\">previous post on nonlinear formulas in linear algebra<\/a>. Let us consider a block matrix \\( {M} \\) of size \\( {(n+m)\\times(n+m)} \\) of the form<\/p>\n<p style=\"text-align: center;\">\\[ M= \\begin{pmatrix} A & B \\\\C & D \\end{pmatrix} \\]<\/p>\n<p style=\"text-align: justify;\">where \\( {A,B,C,D} \\) are \\( {n\\times n} \\), \\( {n\\times m} \\), \\( {m\\times n} \\), \\( {m\\times m} \\). If \\( {D} \\) is invertible then<\/p>\n<p style=\"text-align: center;\">\\[ \\det(M)=\\det(A-BD^{-1}C)\\det(D). \\]<\/p>\n<p style=\"text-align: justify;\">This follows immediately from the identity (mentioned in <a href=\"http:\/\/en.wikipedia.org\/wiki\/Determinant#Block_matrices\">Wikipedia<\/a>)<\/p>\n<p style=\"text-align: center;\">\\[ \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\begin{pmatrix} I & 0 \\\\ -D^{-1}C & I \\end{pmatrix} = \\begin{pmatrix} A-BD^{-1}C & B \\\\ 0 & D \\end{pmatrix} \\]<\/p>\n<p style=\"text-align: justify;\">(the determinant of a block triangular matrix is the product of the determinants of its diagonal blocks). If \\( {m=n} \\) and if \\( {C},{D} \\) commute then \\( {\\det(M)=\\det(AD-BC)} \\). Note that \\( {A-BD^{-1}C} \\) is the Schur complement of \\( {A} \\) in \\( {M} \\). Similar formulas are derived in <a href=\"http:\/\/arxiv.org\/abs\/1112.4379\">arXiv:1112.4379<\/a> for the determinant of \\( {nN\\times nN} \\) block matrices formed by \\( {N^2} \\) blocks of size \\( {n\\times n} \\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a tiny followup of a previous post on nonlinear formulas in linear algebra. Let us consider a block matrix \\( {M} \\) of&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2012\/10\/14\/determinant-of-block-matrices\/\">Continue reading<span class=\"screen-reader-text\">Determinant of block matrices<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":12767},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5214"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=5214"}],"version-history":[{"count":8,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5214\/revisions"}],"predecessor-version":[{"id":5222,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5214\/revisions\/5222"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=5214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=5214"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=5214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}