{"id":15116,"date":"2022-08-17T10:45:23","date_gmt":"2022-08-17T02:45:23","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=15116"},"modified":"2026-03-23T11:24:50","modified_gmt":"2026-03-23T03:24:50","slug":"lia002_5-3","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e7%b7%9a%e6%80%a7%e4%bb%a3%e6%95%b8%e4%ba%8c-linear-algebra-ii-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e8%8e%8a%e9%87%8d%e8%80%81%e5%b8%ab\/lia002_5-3","title":{"rendered":"5.3 Matrix Limits and Markov Chains"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"15116\" class=\"elementor elementor-15116\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section elementor-top-section elementor-element elementor-element-32d13d59 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" 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href=\"https:\/\/ocw.nycu.edu.tw\/\">Homepage<\/a><\/span><\/span><\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-top-section elementor-element elementor-element-491fafd4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"491fafd4\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-1486afe2\" data-id=\"1486afe2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-48bb3b9d elementor-widget elementor-widget-penci-info-box\" data-id=\"48bb3b9d\" data-element_type=\"widget\" data-widget_type=\"penci-info-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"penci_info_box_24289\" class=\"penci-block-vc penci-info-box penci-ibox-float-left penci-view-default penci-shape-circle\">\r\n\t\t\t<div class=\"penci-ibox-inner\">\r\n\t\t\t\t\t\t\t\t<div class=\"penci-ibox-icon penci-ibox-icon--icon penci-icon penci-tibox-text\"><span class=\"penci-ibox-icon-fa\"><\/span><\/div>\t\t\t\t<div class=\"penci-ibox-content-wrap\">\r\n\t\t\t\t\t\t\t\t\t\t<h3 class=\"penci-ibox-title\">5.3 Matrix Limits and Markov Chains\n<\/h3>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ab15b78 elementor-widget elementor-widget-video\" data-id=\"ab15b78\" data-element_type=\"widget\" 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.elementor-wrapper video{position:absolute;top:0;right:0;bottom:0;left:0}}.elementor-widget-video .elementor-open-inline .elementor-custom-embed-image-overlay{position:absolute;top:0;left:0;width:100%;height:100%;background-size:cover;background-position:50%}.elementor-widget-video .elementor-custom-embed-image-overlay{cursor:pointer;text-align:center}.elementor-widget-video .elementor-custom-embed-image-overlay:hover .elementor-custom-embed-play i{opacity:1}.elementor-widget-video .elementor-custom-embed-image-overlay img{display:block;width:100%}.elementor-widget-video .e-hosted-video .elementor-video{-o-object-fit:cover;object-fit:cover}.e-con-inner>.elementor-widget-video,.e-con>.elementor-widget-video{width:var(--container-widget-width);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-inner-section elementor-element elementor-element-3fbd2357 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3fbd2357\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-27e3e82e\" data-id=\"27e3e82e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4ca4de23 elementor-widget elementor-widget-penci-button\" data-id=\"4ca4de23\" data-element_type=\"widget\" data-widget_type=\"penci-button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pcbtn-wrapper pcbtn-align-left\">\r\n\t\t\t<a class=\"pcbtn pcbtn-icon-left elementor-animation-grow\" href=\"\/?post_type=course_page&#038;p=15052\">\r\n\t\t\t\t<span class=\"pcbtn-wrapperin\">\r\n\t\t\t\t\t<span class=\"pcbtn-content\"><svg class=\"e-font-icon-svg e-fas-arrow-left\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M257.5 445.1l-22.2 22.2c-9.4 9.4-24.6 9.4-33.9 0L7 273c-9.4-9.4-9.4-24.6 0-33.9L201.4 44.7c9.4-9.4 24.6-9.4 33.9 0l22.2 22.2c9.5 9.5 9.3 25-.4 34.3L136.6 216H424c13.3 0 24 10.7 24 24v32c0 13.3-10.7 24-24 24H136.6l120.5 114.8c9.8 9.3 10 24.8.4 34.3z\"><\/path><\/svg>\u8fd4\u56de\u8ab2\u7a0b\u9996\u9801<\/span>\r\n\t\t\t\t\t\t\t\t\t<\/span>\r\n\t\t\t<\/a>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-2 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-76f39e5f\" data-id=\"76f39e5f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<div class=\"elementor-element elementor-element-60683f8d elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"60683f8d\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-divider{--divider-border-style:none;--divider-border-width:1px;--divider-color:#0c0d0e;--divider-icon-size:20px;--divider-element-spacing:10px;--divider-pattern-height:24px;--divider-pattern-size:20px;--divider-pattern-url:none;--divider-pattern-repeat:repeat-x}.elementor-widget-divider .elementor-divider{display:flex}.elementor-widget-divider .elementor-divider__text{font-size:15px;line-height:1;max-width:95%}.elementor-widget-divider .elementor-divider__element{margin:0 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100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3b52a3bf elementor-widget elementor-widget-shortcode\" data-id=\"3b52a3bf\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-26\" class=\"tablepress tablepress-id-26\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">\u9031\u6b21<\/th><th class=\"column-2\">\u8ab2\u7a0b\u5167\u5bb9<\/th><th class=\"column-3\">\u8ab2\u7a0b\u5f71\u97f3<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u7b2c\u4e94\u7ae0 Diagonalization<br \/>\n5.1 Eigenvalues and Eigenvectors<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15100\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">5.2 Diagonalizability<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15110\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">5.3 Matrix Limits and Markov Chains<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15116\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">5.4 Invariant Subspaces and The Cayley-Hamiltion Theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15122\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u7b2c\u516d\u7ae0 Inner Product Spaces<br \/>\n6.1 Inner Products and Norms F = R or C<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15128\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6.2 The Gram-Schmidt Orthogonalozation Process and Orthogonal Complements<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15134\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6.3 The Adjoint of a Linear Operator<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15140\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6.5 Unitary and Orthogonal Operators and Their Matrices<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15146\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6.5 Unitary and Orthogonal Operators and Their Matrices<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15152\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6.6 Orthogonal Projections and The Spectral Theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15158\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6.7 The Singular Value Decomposition<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15168\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u7b2c\u4e03\u7ae0 Canonical Forms<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15174\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-26 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-172e9392\" data-id=\"172e9392\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-33ff7026 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