{"id":5746,"date":"2020-11-09T15:44:10","date_gmt":"2020-11-09T06:44:10","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5746"},"modified":"2020-11-23T15:12:11","modified_gmt":"2020-11-23T06:12:11","slug":"linear-algebra-characteristic-subspaces-and-jordan-normal-form","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-characteristic-subspaces-and-jordan-normal-form\/","title":{"rendered":"\ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uacfc \uc870\ub974\ub2f9 \ud45c\uc900\ud615"},"content":{"rendered":"<p>\uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V\\) \uc704\uc5d0\uc11c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uac00 \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \\(V\\)\uc5d0 \uc5b4\ub5a0\ud55c \uae30\uc800\uac00 \uc8fc\uc5b4\uc84c\ub294\uc9c0\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4. \\(V\\)\uc640 \\(<br \/>\nT\\)\uac00 \uc801\uc808\ud55c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(V\\)\uc758 \uae30\uc800\ub97c \uc801\uc808\ud788 \ud0dd\ud558\uc5ec \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \u2018\ub300\ub2e8\ud788 \uc88b\uc740 \ud615\ud0dc\u2019\uac00 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubca1\ud130\uacf5\uac04\uc744 \ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uacfc \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc744 \uc870\ub974\ub2f9 \ud45c\uc900\ud615\uc73c\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ubca1\ud130\uacf5\uac04\uc740 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc778 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n<div style=\"display: none; visibility: hidden;\">\n\\[<br \/>\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}<br \/>\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}<br \/>\n\\newcommand{\\proj}{{\\operatorname{proj}}}<br \/>\n\\newcommand{\\adj}{{\\operatorname{adj}}}<br \/>\n\\newcommand{\\Ker}{{\\operatorname{Ker}}}<br \/>\n\\]\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uc744 \uc774\uc6a9\ud55c \ubd84\ud574<\/h2>\n<p>\uc774 \uc808\uc5d0\uc11c\ub294 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc5d0 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(T\\)\uc5d0 \ub300\ud55c \ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \\(V\\)\ub97c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p>\uc774 \uc8fc\uc81c\ub97c \ub9c8\uce60 \ub54c\uae4c\uc9c0 \\(V\\)\ub294 \uccb4 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \ucc28\uc6d0\uc774 \\(n\\)\uc778 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uba70 \\(T\\)\ub294 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \\(p(t)\\)\ub294 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \ub610\ud55c \\(p(t)\\)\uac00 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4.<\/p>\n<p>\ub17c\uc758\ub97c \uc704\ud558\uc5ec \ucd94\uc0c1\ub300\uc218\ud559\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \uba87 \uac00\uc9c0 \uc6a9\uc5b4\uc640 \uc815\ub9ac\ub97c \ub3c4\uc785\ud558\uc790. \uc0c1\uc218\uac00 \uc544\ub2cc \ub2e4\ud56d\uc2dd \\(p(t) \\in K[t]\\)\uac00 \u2018\\(K\\)\uc5d0\uc11c <span class=\"defined\">\uae30\uc57d<\/span>(irreducible)\uc774\ub2e4\u2019\ub77c\ub294 \uac83\uc740 \\(g(t)\\)\uac00<br \/>\n\\[g(t) = h_1 (t) h_2 (t) \\quad (h_1 (t) ,\\, h(t) \\in K[t])\\]<br \/>\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0a0 \ub54c\ub294 \\(h_1(t)\\) \ub610\ub294 \\(h_2(t)\\)\uac00 \uc0c1\uc218\uc77c \ub54c\ubfd0\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc989 \\(g(t)\\)\uac00 \uae30\uc57d\ub2e4\ud56d\uc2dd\uc774\uba74 \\(g(t)\\)\ub294 \ucc28\uc218\uac00 \ub354 \ub0ae\uace0 \ubb38\uc790\ub97c \uac00\uc9c4 \ub450 \uac1c\uc758 \ub2e4\ud56d\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub420 \uc218 \uc5c6\ub2e4. \ub3d9\uc77c\ud55c \ub2e4\ud56d\uc2dd\uc774\ub77c\ub3c4 \uccb4 \\(K\\)\uac00 \ubb34\uc5c7\uc778\uc9c0\uc5d0 \ub530\ub77c \uae30\uc57d\ub2e4\ud56d\uc2dd\uc77c \uc218\ub3c4 \uc788\uace0 \uae30\uc57d\uc774 \uc544\ub2d0 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<p>\uc774\uc81c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc99d\uba85 \uc5c6\uc774 \ubc1b\uc544\ub4e4\uc774\uae30\ub85c \ud55c\ub2e4.<\/p>\n<ul>\n<li><span class=\"theorem\">\uc720\uc77c\ubd84\ud574\uc131\uc9c8.<\/span> \\(K[t]\\)\uc5d0 \uc18d\ud558\ub294, \uc0c1\uc218\uac00 \uc544\ub2cc \ub2e4\ud56d\uc2dd\uc740 \\(K[t]\\)\uc5d0\uc11c \uae30\uc57d\uc778 \ub2e4\ud56d\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4. \ub354\uc6b1\uc774, \uc774\ub7ec\ud55c \uc778\uc218\ubd84\ud574 \ud45c\ud604\uc740 \uc778\uc218\uc758 \uc21c\uc11c\uc640 \uc0c1\uc218\ubc30\ub97c \uc81c\uc678\ud558\uba74 \uc720\uc77c\ud55c \uaf34\uc774\ub2e4.<\/li>\n<li><span class=\"theorem\">B\u00e9zout \ud56d\ub4f1\uc2dd.<\/span> \\(g_1,\\) \\(\\cdots,\\) \\(g_m\\)\uc774 \\(K[t]\\)\uc5d0 \uc18d\ud558\ub294 \ub2e4\ud56d\uc2dd\uc774\uace0, \ubb38\uc790\ub97c \ud3ec\ud568\ud55c \uacf5\ud1b5\uc778\uc218\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4\uace0 \ud558\uc790. (\uc989 \uacf5\ud1b5\uc778\uc218\uac00 \uc0c1\uc218 \ubfd0\uc774\ub77c\uace0 \ud558\uc790.) \uadf8\ub7ec\uba74 \\(K[t]\\)\uc5d0 \uc18d\ud558\ub294 \ub2e4\ud56d\uc2dd \\(h_1,\\) \\(\\cdots ,\\) \\(h_m\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<br \/>\n\\[g_1 (t) h_1 (t) + \\cdots + g_m (t) h_m (t) = 1\\]<br \/>\n\uc5ec\uae30\uc11c \uc6b0\ubcc0\uc740 \uc0c1\uc218\ub2e4\ud56d\uc2dd\uc744 \ub098\ud0c0\ub0b8\ub2e4.\n<\/li>\n<\/ul>\n<p>\uc774\uc81c \ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uc5d0 \uad00\ud55c \ub17c\uc758\ub97c \uc774\uc5b4\uac00\uc790. \\(p(t)\\)\uac00 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub418\ubbc0\ub85c \\(\\lambda_1,\\) \\(\\cdots,\\) \\(\\lambda_n \\in K\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[p(t) = \\prod_{j=1}^n (t-\\lambda_j )\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ubb3c\ub860 \ubaa8\ub4e0 \\(\\lambda_j\\) \uc911\uc5d0\ub294 \uc11c\ub85c \uac19\uc740 \uac83\uc774 \uc874\uc7ac\ud560 \uc218 \uc788\ub2e4. \\(\\lambda_j\\)\uac00 \uac16\ub294 \uac12\uc758 \uac1c\uc218\uac00 \\(r\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc704 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[p(t) = \\prod_{j=1}^r (t-\\lambda_j )^{m_j} .\\tag{1}\\]<br \/>\n\uc5ec\uae30\uc11c \\(m_j\\)\ub97c \uace0\uc733\uac12 \\(\\lambda_j\\)\uc758 <span class=\"defined\">\uc911\ubcf5\ub3c4<\/span>(multiplicity)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(j=1,\\, 2,\\, \\cdots,\\, r\\)\uc5d0 \ub300\ud558\uc5ec \\(V\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(N_j\\)\ub97c<br \/>\n\\[N_j = (T-\\lambda_j I_n )^{m_j}\\tag{2}\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud558\uace0, \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04 \\(U_j\\)\ub97c<br \/>\n\\[U_j = \\Ker (N_j )\\tag{3}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uac01 \\(U_j\\)\ub294 \\(\\lambda_j\\)\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\uacf5\uac04\uc744 \ud3ec\ud568\ud558\ubbc0\ub85c \uc790\uba85\ud558\uc9c0 \uc54a\uc740(\uc601\ubca1\ud130 \uc678\uc758 \ubca1\ud130\ub97c \uac00\uc9c0\ub294) \uacf5\uac04\uc774\ub2e4. \uc774\ub85c\uc368 \ub2e4\uc74c \uc815\uc758\ub97c \ub3c4\uc785\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"definition margintop2\">\n<p><span class=\"definition\">\uc815\uc758 1.<\/span><br \/>\n\\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04 \\(U_1,\\) \\(\\cdots,\\) \\(U_r\\)\ub97c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uc5d0 \ub300\ud55c <span class=\"defined\">\ud2b9\uc131\ubd80\ubd84\uacf5\uac04<\/span>(characteristic subspace)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n<p>\uc774 \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \uc2a4\ud399\ud2b8\ub7fc \ubd84\ud574 \uc815\ub9ac\ub97c \uc77c\ubc18\ud654\ud560 \uc218 \uc788\ub2e4. \uc989 \\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\uac70\ub098 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774 \uc544\ub2c8\uace0, \\(K\\)\uac00 \uc2e4\uc218\uccb4\uc774\uac70\ub098 \ubcf5\uc18c\uc218\uccb4\ub77c\ub294 \uac00\uc815 \uc5c6\uc774 \ub17c\uc758\ub97c \uc9c4\ud589\ud560 \uc218 \uc788\ub2e4. \ub2e8\uc9c0 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4\ub294 \uac00\uc815\uc744 \ubc14\ud0d5\uc73c\ub85c \ud558\uace0 \uc788\ub2e4\ub294 \uc810\uc744 \uc5fc\ub450\uc5d0 \ub450\uc790.<\/p>\n<div class=\"theorem margintop2\">\n<p class=\"marginbottom0\"><span class=\"theorem\">\uc815\ub9ac 2.<\/span><br \/>\n\\(U_1,\\) \\(\\cdots,\\) \\(U_r\\)\uac00 \\(T\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 \ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis-roman\">\n<li>\uac01 \ubd80\ubd84\uacf5\uac04 \\(U_j\\)\ub294 \\(T\\)-\ubd88\ubcc0\uc774\ub2e4.<\/li>\n<li>\\(V = U_1 \\oplus \\cdots \\oplus U_r .\\)<\/li>\n<li>\uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec, \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(U_j\\)\ub85c \uc81c\ud55c\ud55c \ud568\uc218 \\(T\\vert_{U_j}\\)\ub294 \ub2e8 \ud558\ub098\uc758 \uace0\uc733\uac12 \\(\\lambda_j\\)\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<li>\uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec, \\(U_j\\)\uc758 \ucc28\uc6d0\uc740 \\(\\lambda_j\\)\uc758 \uc911\ubcf5\ub3c4\uc640 \uac19\ub2e4. \uc989 \\(\\dim(U_j) = m_j\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>(i) \uc815\uc758 (2)\uc5d0 \uc758\ud558\uc5ec \uac01 \\(N_j\\)\ub294 \\(T\\)\uc640 \uad50\ud658\ud560 \uc218 \uc788\ub2e4. \ub610\ud55c \\(N_j (U_j) = \\left\\{ \\mathbf{0} \\right\\}\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[N_j T(U_j) = TN_j (U_j ) = T(\\left\\{ \\mathbf{0} \\right\\}) = \\left\\{ \\mathbf{0} \\right\\}\\]<br \/>\n\uc774\ubbc0\ub85c, \\(T(U_j)\\)\ub294 \\(\\Ker(N_j) = U_j\\)\uc5d0 \ud3ec\ud568\ub41c\ub2e4.<\/p>\n<p>(ii) \uba3c\uc800 \\(V = U_1 + \\cdots + U_r\\)\uc744 \ubcf4\uc774\uc790. \ub2e4\uc74c\uacfc \uac19\uc740 \ub2e4\ud56d\uc2dd\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[q_j (t) = \\frac{p(t)}{(t-\\lambda_j)^{m_k}} . \\quad (j=1,\\,2,\\,\\cdots,\\,r)\\]<br \/>\n\uc989 \\(p(t)\\)\uc758 \uc778\uc218 \uc911\uc5d0\uc11c \\(\\lambda_j\\)\ub97c \uadfc\uc73c\ub85c \uac16\ub294 \uc778\uc218\ub97c \uc81c\uac70\ud55c \uac83\uc774\ub2e4. Cayley-Hamilton \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[[N_j q_j (T) ](v) = p(T) (v) = \\mathbf{0}\\]<br \/>\n\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(q_j (T) (v) \\in U_j\\)\uc774\ub2e4. \ub354\uc6b1\uc774, \\(q_j\\)\ub4e4\uc740 \uc0c1\uc218 \uc678\uc758 \uacf5\ud1b5\uc778\uc218\ub97c \uac16\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \ub2e4\ud56d\uc2dd \\(h_1 (t) ,\\) \\(\\cdots,\\) \\(h_r(t)\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[h_1 (t) q_1 (t) + \\cdots + h_r (t) q_r (t) = 1\\]<br \/>\n\uc989<br \/>\n\\[h_1 (T) q_1 (T) + \\cdots h_r (T) q_r (T) = 1_V \\tag{4}\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[v = \\sum_{j=1}^r [ h_j (T) q_j (T)](v)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(q_j (T)(v)\\in U_j\\)\uc774\uace0 \\(U_j\\)\uac00 \\(T\\)-\ubd88\ubcc0\uc774\ubbc0\ub85c, \uc704 \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc758 \ud569\uc5d0\uc11c \\(j\\)\uc9f8 \ud56d\uc740 \\(U_j\\)\uc5d0 \uc18d\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(V = U_1 + \\cdots + U_r\\)\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(U_1 \\cap \\cdots \\cap U_r = \\left\\{ \\mathbf{0} \\right\\}\\)\uc744 \uc99d\uba85\ud558\uc790. \uac01 \\(u_k \\in U_k\\) \\((k=1,\\,2,\\,\\cdots,\\,r)\\)\uc640 \\(j=1,\\,2,\\,\\cdots,\\,r\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[[h_j (T) q_j (T) ](u_k) =<br \/>\n\\begin{cases}<br \/>\nu_k &#038;\\quad \\text{ if } j=k \\\\[5pt]<br \/>\n\\mathbf{0} &#038;\\quad \\text{ otherwise}<br \/>\n\\end{cases}\\tag{5}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\uac83\uc744 \ubcf4\uc774\uae30 \uc704\ud558\uc5ec \\(j\\ne k\\)\uc77c \ub54c \\(q_j\\)\uac00 \uc778\uc218 \\(N_k\\)\ub97c \uac00\uc9c0\uba70 \uac01 \\(U_k = \\Ker (N_k)\\)\uc5d0\uc11c \\(q_j\\)\uc758 \ud568\uc22b\uac12\uc740 \\(\\mathbf{0}\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (5)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774 \uc0ac\uc2e4\uacfc (4)\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[u_k = \\sum_{j=1}^r [ h_j (T) q_j (T) ](u_k) = [ h_k (T) q_k (T)] (u_k).\\]<br \/>\n\uc774\uc81c \\(u_j \\in U_j\\)\uc774\uace0<br \/>\n\\[u_1 + \\cdots + u_r = \\mathbf{0}\\tag{6}\\]<br \/>\n\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec (6)\uc758 \uc591\ubcc0\uc5d0 \\(h_j (T) q_j (T)\\)\ub97c \ucde8\ud558\uba74 (5)\uc5d0 \uc758\ud558\uc5ec \\(u_j = \\mathbf{0}\\)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p><p>(iii) \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(U_j\\)\ub85c \uc81c\ud55c\ud55c \ud568\uc218\ub97c \\(T_j = T\\vert_{U_j}\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub9ac\uace0 \\(T_j\\)\uc758 \uace0\uc733\uac12\uc744 \\(\\mu_j\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\((\\mu_j &#8211; \\lambda_j )^{m_j}\\)\ub294 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\((T_j &#8211; \\lambda_j 1_{U_j} )^{m_j}\\)\uc758 \uace0\uc733\uac12\uc774\ub2e4. \uadf8\ub7f0\ub370  \\((T_j &#8211; \\lambda_j 1_{U_j} )^{m_j}\\)\uc740 \\(U_j\\)\uc5d0\uc11c \ud568\uc22b\uac12 \\(\\mathbf{0}\\)\uc744 \ucde8\ud558\ubbc0\ub85c, \uc774 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \uace0\uc733\uac12\uc740 \\(0\\) \ubfd0\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mu_j = \\lambda_j\\)\uc774\ub2e4. [\uc2a4\uce7c\ub77c \\(\\lambda_j\\)\ub294 \uadf8 \uc790\uccb4\ub85c \uc81c\ud55c\uc0ac\uc0c1\uc758 \uace0\uc733\uac12\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(U_j\\)\ub294 \\(\\lambda_j\\)\uc5d0 \ub300\uc751\ud558\ub294 \uace0\uc720\uacf5\uac04 \uc804\uccb4\ub97c \ud3ec\ud568\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.]<\/p>\n<p>(iv) \uac01 \\(j=1,\\,2,\\,\\cdots,\\,r\\)\uc5d0 \ub300\ud558\uc5ec \\(n_j = \\dim(U_j)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(B_j\\)\uac00 \\(U_j\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(B_1 ,\\) \\(\\cdots ,\\) \\(B_r\\)\ub97c \uc774 \uc21c\uc11c\ub300\ub85c \ud569\uc9d1\ud569\ud558\uc5ec \\(V\\)\uc758 \uae30\uc800 \\(B\\)\ub97c \uc5bb\uc73c\uba70, \uc774 \uae30\uc800\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\uc774 \ub41c\ub2e4.<br \/>\n\\[A =<br \/>\n\\left(\\begin{array}{cccc}<br \/>\nA_1 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; A_2 &#038; \\cdots &#038; 0 \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\n0 &#038; 0 &#038; \\cdots &#038; A_r<br \/>\n\\end{array}\\right).\\]<br \/>\n\uc5ec\uae30\uc11c \uac01 \\(A_j\\)\ub294 \uae30\uc800 \\(B_j\\)\uc5d0 \ub300\ud55c \\(T_j\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774\uba70, \\(n_j \\times n_j\\) \ud589\ub82c\uc774\ub2e4. \ub530\ub77c\uc11c \\(V\\)\uc5d0\uc11c \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \\(T_j\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\ub4e4\uc758 \uacf1\uacfc \uac19\ub2e4. (iii)\uc5d0 \uc758\ud558\uc5ec \uac01 \\(T_j\\)\ub294 \\(\\lambda_j\\)\ub9cc\uc744 \uace0\uc733\uac12\uc73c\ub85c \uac00\uc9c0\ubbc0\ub85c \\(T_j\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \\((t-\\lambda_j)^{n_j}\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (1)\uc5d0 \uc758\ud558\uc5ec \\(V\\)\uc5d0\uc11c \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740<br \/>\n\\[\\prod_{j=1}^r (t-\\lambda_j )^{m_j} = \\prod_{j=1}^r (t-\\lambda_j )^{n_j}\\]<br \/>\n\uc774\ub2e4. \ub2e4\ud56d\uc2dd\uc758 \uc720\uc77c\ubd84\ud574\uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \uac01 \\(j=1,\\,\\cdots,\\,r\\)\uc5d0 \ub300\ud558\uc5ec \\(m_j = n_j\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\uc870\ub974\ub2f9 \ud45c\uc900\ud615<\/h2>\n<p>\uc774 \uc808\uc5d0\uc11c\ub294 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \uc0bc\uac01\ud589\ub82c \ucd95\uc57d \uc815\ub9ac\ub97c \uc77c\ubc18\ud654\ud55c \ub0b4\uc6a9\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\uc774 \uc8fc\uc81c\ub97c \ub9c8\uce60 \ub54c\uae4c\uc9c0 \uc0bc\uac01\ud589\ub82c\uc740 \uc704 \uc0bc\uac01\ud589\ub82c\uc744 \uc774\ub974\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \ub9cc\uc57d \uc0bc\uac01\ud589\ub82c\uc758 \ub300\uac01\uc131\ubd84\uc774 \ubaa8\ub450 \\(0\\)\uc774\uba74, \uadf8 \uc0bc\uac01\ud589\ub82c\uc744 <span class=\"defined\">\uc21c\uc0bc\uac01\ud589\ub82c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc21c\uc0bc\uac01\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 3.<\/span><br \/>\n\\(A\\in M_n (K)\\)\uac00 \uc21c\uc0bc\uac01\ud589\ub82c\uc774\uba74 \\(A^n = O\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(n\\)\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc73c\ub85c \uc27d\uac8c \uc99d\uba85\ub41c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(A\\)\uac00 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uace0 \\(A^m = O\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud560 \ub54c, \\(A\\)\ub97c <span class=\"defined\">\uba71\uc601\uc6d0<\/span>(nilpotent)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(A^m = O\\)\uc774 \ub418\ub294 \uc790\uc5f0\uc218 \\(m\\) \uc911\uc5d0\uc11c \uac00\uc7a5 \uc791\uc740 \uac12\uc744 \\(A\\)\uc758 <span class=\"defined\">\uba71\uc601\uc9c0\uc218<\/span>(index of nilpotency)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub7ec\ud55c \uc6a9\uc5b4\ub294 \ubca1\ud130\uacf5\uac04 \uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc5d0\ub3c4 \ub611\uac19\uc774 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \uc815\uc758\uc5ed\uc744 \ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uc73c\ub85c \uc81c\ud55c\ud55c \ud568\uc218\uc758 \uc131\uc9c8\uc744 \ub354 \uae4a\uc774 \uc0b4\ud3b4\ubcf4\uc790. \\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(n\\)\ucc28\uc6d0\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(T\\)\uac00 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4\uace0 \ud558\uc790. \uadf8 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc758 \uadfc\uc744 \\(\\lambda_1,\\) \\(\\cdots,\\) \\(\\lambda_r\\)\ub77c\uace0 \ud558\uace0, \uc774 \uadfc\ub4e4\uc758 \uc911\ubcf5\ub3c4\ub97c \\(m_1,\\) \\(\\cdots,\\) \\(m_r\\)\ub77c\uace0 \ud558\uba70, \uc774 \uadfc\ub4e4\uc5d0 \ub300\uc751\ub418\ub294 \ud2b9\uc131\uacf5\uac04\uc744 \\(U_1 ,\\) \\(\\cdots ,\\) \\(U_r\\)\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(U_j\\)\ub85c \uc81c\ud55c\ud55c \ud568\uc218\ub97c \\(T_j : U_j \\rightarrow U_j\\)\ub85c \ub098\ud0c0\ub0b4\uc790.<\/p>\n<p>\uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \uc0bc\uac01\ud589\ub82c \ucd95\uc57d \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uac01 \\(U_j\\)\uc758 \uae30\uc800 \\(B_j\\)\uac00 \uc874\uc7ac\ud558\uba70, \uc774 \uae30\uc800\uc5d0 \ub300\ud55c \\(T_j\\)\uc758 \ud45c\ud604\ud589\ub82c \\(A_j\\)\uac00 \uc0bc\uac01\ud589\ub82c\uc774 \ub41c\ub2e4. \uc0bc\uac01\ud589\ub839\uc758 \uace0\uc733\uac12\uc740 \ub300\uac01\uc131\ubd84\uacfc \uac19\uc740\ub370, \uc815\ub9ac 2\uc758 (iii)\uc5d0 \uc758\ud558\uc5ec \\(T_j\\)\ub294 \\(\\lambda_j\\)\ub9cc\uc744 \uace0\uc733\uac12\uc73c\ub85c \uac00\uc9c0\ubbc0\ub85c \ud589\ub82c \\(A_j\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\uc774 \ub41c\ub2e4.<br \/>\n\\[A = \\left(\\begin{array}{ccccc}<br \/>\n\\lambda_j &#038; \\ast &#038; \\cdots &#038; \\ast &#038; \\ast \\\\<br \/>\n0 &#038; \\lambda_j &#038; \\cdots &#038; \\ast &#038; \\ast \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots &#038; \\vdots \\\\<br \/>\n0 &#038; 0 &#038; \\cdots &#038; \\lambda_j &#038; \\ast \\\\<br \/>\n0 &#038; 0 &#038; \\cdots &#038; 0 &#038; \\lambda_j<br \/>\n\\end{array}\\right).\\]<br \/>\n\uba85\ubc31\ud788 \\(A_j\\)\ub294 \uc2a4\uce7c\ub77c\ud56d\ub4f1\ud589\ub82c \\(\\lambda_j I_{m_j}\\)\uc640 \uc21c\uc0bc\uac01\ud589\ub82c\uc758 \ud569\uc73c\ub85c \ud45c\ud604\ub420 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uacfc \uac19\uc740 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.<\/span><br \/>\n\uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(U_j\\)\ub85c \uc81c\ud55c\ud55c \ud568\uc218 \\(T_j\\)\ub294 \ub300\uac01\ud589\ub82c \\(D_j = \\lambda_j I_{m_j}\\)\uc640 \uba71\uc601\ud589\ub82c \\(N_j\\)\uc758 \ud569<br \/>\n\\[T_j = D_j + N_j\\tag{7}\\]<br \/>\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n<p>(7)\uc5d0\uc11c \\(D_j\\)\ub97c \\(T_j\\)\uc758 <span class=\"defined\">\ub300\uac01\ubd80\ubd84<\/span>(diagonal part)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(N_j\\)\ub97c \\(T_j\\)\uc758 <span class=\"defined\">\uba71\uc601\ubd80\ubd84<\/span>(nilpotent part)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uac01 \\(T_j\\)\uc5d0 \ub300\ud558\uc5ec \\(D_j\\)\uc640 \\(N_j\\)\ub294 \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c4\ub2e4.<\/p>\n<p>\\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(n\\)\ucc28\uc6d0\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(N\\)\uc774 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uba71\uc601\ubcc0\ud658\uc774\uba70 \\(N^m\\)\uc774 \uc601\ubcc0\ud658(\ubaa8\ub4e0 \uac12\uc744 \uc601\ubca1\ud130\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ubcc0\ud658)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(V\\)\uc5d0\uc11c\uc758 \uc801\ub2f9\ud55c \uae30\uc800\ub97c \uad6c\uc131\ud558\uc5ec, \uadf8 \uae30\uc800\uc5d0 \ub300\ud558\uc5ec \\(N\\)\uc774 \uac04\ub2e8\ud55c \ud589\ub82c\ub85c \ud45c\ud604\ub420 \uc218 \uc788\ub3c4\ub85d \ud574\ubcf4\uc790.<\/p>\n<p>\uba3c\uc800 \uba87 \uac1c\uc758 \ubcf4\uc870\uc815\ub9ac\uac00 \ud544\uc694\ud558\ub2e4.<\/p>\n<div class=\"lemma margintop2\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 5.<\/span><br \/>\n\\(w_0\\in V,\\) \\(w_0 \\ne \\mathbf{0}\\)\uc774\uace0 \\(m\\)\uc774 \\(N^m (w_0 ) = \\mathbf{0}\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac00\uc7a5 \uc791\uc740 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubca1\ud130<br \/>\n\\[N^{m-1} (w_0) ,\\,\\, N^{m-2} (w_0) ,\\,\\, \\cdots,\\,\\, N(w_0) ,\\,\\, w_0\\tag{8}\\]<br \/>\n\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uc8fc\uc5b4\uc9c4 \ubca1\ud130\ub4e4\uc774 \uc77c\ucc28\ub3c5\ub9bd\uc784\uc744 \ubcf4\uc774\uae30 \uc704\ud558\uc5ec<br \/>\n\\[\\sum_{j=1}^m \\mu_j N^{m-j} (w_0) = \\mathbf{0}\\]<br \/>\n\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(N^{m-1}\\)\uc744 \ucde8\ud558\uba74, \uc88c\ubcc0\uc758 \ud569 \uc911\uc5d0\uc11c \ub9c8\uc9c0\ub9c9 \ud56d\uc744 \uc81c\uc678\ud55c \ubaa8\ub4e0 \ud56d\uc740 \\(\\mathbf{0}\\)\uc774 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mu_m = 0\\)\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(N^{m-2}\\)\uc744 \ucde8\ud558\uba74, \uc88c\ubcc0\uc758 \ud569 \uc911\uc5d0\uc11c \ub9c8\uc9c0\ub9c9\uc5d0\uc11c \ub450 \ubc88\uc9f8 \ud56d\uc744 \uc81c\uc678\ud55c \ubaa8\ub4e0 \ud56d\uc740 \\(\\mathbf{0}\\)\uc774 \ub418\ubbc0\ub85c \\(\\mu_{m-1}=0\\)\uc774\ub2e4. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec, \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc758 \ubaa8\ub4e0 \uacc4\uc218\uac00 \\(0\\)\uc784\uc744 \uc54c \uc218 \uc788\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>(8)\uc758 \ubca1\ud130\ub4e4\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \ubd80\ubd84\uacf5\uac04 \\(W\\)\ub97c \\(N\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 <span class=\"defined\">\uc21c\ud658\ubd80\ubd84\uacf5\uac04<\/span>(cyclic subspace)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub54c (8)\uc744 \\(W\\)\uc758 <span class=\"defined\">\uc21c\ud658\uae30\uc800<\/span>(cyclic basis)\ub77c\uace0 \ubd80\ub974\uba70, \\(w_0\\)\ub97c \uc774 \uae30\uc800\uc758 <span class=\"defined\">\uadfc<\/span>(root)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uba85\ubc31\ud788 \uc774\uc640 \uac19\uc740 \ubd80\ubd84\uacf5\uac04\uc740 \\(N\\)-\ubd88\ubcc0\uc774\uba70, \uae30\uc800 (8)\uc5d0 \ub300\ud55c \\(N\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \\(m\\times m\\) \ud589\ub82c\uc774\ub2e4.<br \/>\n\\[\\left(\\begin{array}{ccccc}<br \/>\n0 &#038; 1 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; \\cdots &#038; 0<br \/>\n\\end{array}\\right)\\]<br \/>\n\uacf5\uac04 \\(V\\) \uc790\uccb4\ub294 \uc21c\ud658\uacf5\uac04\uc774 \uc544\ub2d0 \uc218\ub3c4 \uc788\ub2e4. \uc989 \\(N^j (v_0),\\) \\(j=0,\\,1,\\,\\cdots,\\,n-1\\)\uc774 \\(V\\)\uc758 \uae30\uc800\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \ubca1\ud130 \\(v_0\\)\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uc6b0\ub9ac\ub294 \\(V\\)\uc758 \uc801\ub2f9\ud55c \uae30\uc800 \\(B\\)\ub97c \uad6c\uc131\ud558\uc5ec, \\(B\\)\uc5d0 \ub300\ud55c \\(N\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \uac19\uc740 \uc131\uc9c8\uc744 \uac16\ub3c4\ub85d \ud560 \uac83\uc774\ub2e4. \ud575\uc2ec\uc740 \uacf5\uac04 \\(V\\)\ub97c \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ud45c\ud604\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<div class=\"lemma margintop2\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 6.<\/span><br \/>\n\\(W\\)\uc758 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc774\uace0 \ucc28\uc6d0\uc774 \\(n\\)\uc774\uba70 \uc21c\ud658\uae30\uc800<br \/>\n\\[N^{m-1}(w_0) ,\\,\\, N^{m-2}(w_0) ,\\,\\, \\cdots ,\\,\\, N(w_0) ,\\,\\, w_0\\]<br \/>\n\uc744 \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(W\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(w\\)\uc5d0 \ub300\ud558\uc5ec \uc801\ub2f9\ud55c \\(q(t)\\in K[t]\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(w= q(N) (w_0)\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(w\\in W\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(w\\)\ub294<br \/>\n\\[w = \\sum_{j=1}^m \\mu_j N^{m-j} (w_0)\\]<br \/>\n\uc758 \uaf34\ub85c \ud45c\ud604\ub41c\ub2e4.<br \/>\n\\[q(t) = \\sum_{j=1}^m \\mu_j t^{m-j}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uba74 \\(w=q(N)(w_0)\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"lemma margintop2\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 7.<\/span><br \/>\n\\(W\\)\uac00 \ubcf4\uc870\uc815\ub9ac 6\uc5d0\uc11c\uc640 \uac19\uc740 \uacf5\uac04\uc774\uace0, \uc801\ub2f9\ud55c \\(q(t)\\in K[t]\\)\uc5d0 \ub300\ud558\uc5ec \\(q(N)\\)\uc774 \\(W\\) \uc704\uc5d0\uc11c \uc601\ubcc0\ud658\uc774 \ub41c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(q(t)\\)\ub294 \\(t^m\\)\uc744 \uc778\uc218\ub85c \uac00\uc9c4\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\ub098\ub217\uc148 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc801\ub2f9\ud55c \\(q_1 (t) ,\\, r(t) \\in K[t]\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[q(t) = q_1 (t) t^m + r(t)\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc5ec\uae30\uc11c \\(r(t)\\)\uc758 \ucc28\uc218\ub294 \\(m\\) \ubbf8\ub9cc\uc774\ub2e4 \\(N^m\\)\uc740 \\(W\\) \uc704\uc5d0\uc11c \uc601\ubcc0\ud658\uc774\ubbc0\ub85c, \\(W\\)\uc5d0 \uc8fc\uc5b4\uc9c4 \uc21c\ud658\uae30\uc800\uc758 \uadfc \\(w_0\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[q(N) (w_0) = [q_1 (N) N^m ] (w_0) + r(N) (w_0) =r(N) (w_0) \\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c, \ub9cc\uc57d \\(q(N)\\)\uc774 \\(W\\) \uc704\uc5d0\uc11c \uc601\ubcc0\ud658\uc774\uba74 \\(r(N) (w_0) = \\mathbf{0}\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(r\\)\uac00 \uc601\ub2e4\ud56d\uc2dd\uc774 \uc544\ub2c8\uba74 \uc21c\ud658\uae30\uc800\uc758 \uc77c\ucc28\ub3c5\ub9bd\uc131\uc5d0 \uc758\ud558\uc5ec \uc774 \ub4f1\uc2dd\uc740 \uc131\ub9bd\ud560 \uc218 \uc5c6\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(q(t) = q_1 (t) t^m\\)\uc774\uba70, \\(t^m\\)\uc740 \\(q(t)\\)\uc758 \uc778\uc218\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"lemma margintop2\">\n<p class=\"marginbottom0\"><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 8.<\/span><br \/>\n\\(N\\)\uc774 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \uba71\uc601\ubcc0\ud658\uc774\uba70, \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04 \\(W\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis-roman marginbottom0\">\n<li>\\(N(W) = N(V).\\) \uc989 \\(N\\)\uc5d0 \uc758\ud55c \\(W\\)\uc758 \uc0c1\uacfc \\(V\\)\uc758 \uc0c1\uc774 \ub3d9\uc77c\ud558\ub2e4.<\/li>\n<li>\\(W\\)\uac00 \\(N\\)\uc5d0 \ub300\ud55c \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4.<\/li>\n<\/ol>\n<p>\uadf8\ub7ec\uba74 \\(V\\)\ub294 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(v\\in V\\)\ub77c\uace0 \ud558\uc790. \uc870\uac74 (i)\uc5d0 \uc758\ud558\uc5ec \\(w\\in W\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(N(v) = N(w)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774\ub54c \\(v-w \\in \\Ker (N)\\)\uc774\ubbc0\ub85c \\(V = W + \\Ker (N)\\)\uc774\ub2e4. \\(W\\)\uc640 \\(\\Ker(N)\\)\uc758 \uae30\uc800\ub97c \uac01\uac01 \\(B_W,\\) \\(B_K\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(B_K\\)\uc758 \ubd80\ubd84\uacf5\uac04 \\(B_K &#8216; \\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(B_W \\cup B_K &#8216; \\)\uc774 \\(V\\)\uc758 \uae30\uc800\uac00 \ub41c\ub2e4. \\(B_K &#8216; \\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \uacf5\uac04\uc744 \\(W &#8216; \\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V = W \\oplus W &#8216; \\)\uc774\ub2e4.<\/p>\n<p>\uac00\uc815 (ii)\uc5d0 \uc758\ud558\uc5ec \\(W\\)\ub294 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc774\ub2e4. \\(W &#8216; \\)\uc740 \uc790\uba85\ud55c \uacf5\uac04\uc774\uac70\ub098 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\Ker (N)\\)\uc5d0 \uc18d\ud558\ub294 \ubca1\ud130 \uc911 \uc601\ubca1\ud130\uac00 \uc544\ub2cc \uac83, \ud2b9\ud788 \\(B_K &#8216; \\)\uc758 \uc784\uc758\uc758 \ubca1\ud130\ub294 \uc77c\ucc28\uc6d0 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc744 \uc0dd\uc131\ud558\uba70 \uc790\uae30 \uc790\uc2e0\uc774 \uadf8 \uacf5\uac04\uc758 \uc21c\ud658\uae30\uc800\uac00 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(V = W \\oplus W &#8216; \\)\uc740 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774\uc81c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 9.<\/span><br \/>\n\\(V\\)\uac00 \uc790\uba85\ud558\uc9c0 \uc54a\uc740 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(N\\)\uc774 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uba70 \uba71\uc601\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(N\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 \uc21c\ud658\ubd80\ubd84\uacf5\uac04 \\(V_1 ,\\) \\(\\cdots,\\) \\(V_s\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[V = V_1 \\oplus V_2 \\oplus \\cdots \\oplus V_s\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc989 \\(V\\)\ub294 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(V\\)\uc758 \ucc28\uc6d0 \\(n\\)\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790.<\/p>\n<p>\\(n=1\\)\uc778 \uacbd\uc6b0 \\(1\\)\ucc28\uc6d0 \uacf5\uac04 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uba71\uc601\ubcc0\ud658\uc740 \uc601\ubcc0\ud658 \ubfd0\uc774\ubbc0\ub85c, \uc601\ubca1\ud130\uac00 \uc544\ub2cc \uc784\uc758\uc758 \ubca1\ud130\ub294 \uadf8 \uacf5\uac04\uc758 \uc21c\ud658\uae30\uc800\uac00 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(n=1\\)\uc77c \ub54c \uc815\ub9ac\uc758 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\\(n > 1\\)\uc774\ub77c\uace0 \ud558\uc790. \uc9c0\uae08\ubd80\ud130 \ubcf4\uc870\uc815\ub9ac\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubd80\ubd84\uacf5\uac04\uc744 \uad6c\uc131\ud560 \uac83\uc774\ub2e4.<\/p>\n<p>\uba3c\uc800 \\(N(V) \\ne V\\)\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(N(V) = V\\)\ub77c\uba74 \uc784\uc758\uc758 \\(m\\)\uc5d0 \ub300\ud558\uc5ec \\(N^m (V)=V\\)\uc774\ubbc0\ub85c \\(N\\)\uc774 \uba71\uc601\ubcc0\ud658\uc774\ub77c\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \ub530\ub77c\uc11c \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(N(V)\\)\ub294 \uc801\ub2f9\ud55c \uc21c\ud658\ubd80\ubd84\uacf5\uac04 \\(W_1,\\) \\(\\cdots,\\) \\(W_r\\)\uc758 \uc9c1\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \uac01 \\(j=1,\\, \\cdots ,\\, r\\)\uc5d0 \ub300\ud558\uc5ec, \\(W_j\\)\uc758 \uc21c\ud658\uae30\uc800\uc758 \uadfc\uc744 \\(w_j\\in W_j\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(N\\)\uc5d0 \uc758\ud55c \\(w_1 ,\\) \\(\\cdots,\\) \\(w_r\\)\uc758 \uc5ed\uc0c1\uc744 \uc21c\uc11c\ub300\ub85c \\(v_1,\\) \\(\\cdots,\\) \\(v_r\\)\ub77c\uace0 \ud558\uc790. \uc989 \\(N(v_j) = w_j\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(W_j \\cup \\left\\{ v_j \\right\\}\\)\uac00 \uc0dd\uc131\ud558\ub294 \uacf5\uac04\uc744 \\(W_j &#8216; \\)\uc774\ub77c\uace0 \ud558\uace0, \\(W &#8216; = W_1 &#8216; + \\cdots + W_r &#8216; \\)\uc774\ub77c\uace0 \uc815\uc758\ud558\uc790. \uba85\ubc31\ud788 \uac01 \\(W_j &#8216; \\)\uc740 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc774\uba70, \\(v_j\\)\ub294 \uc21c\ud658\uae30\uc800\uc758 \uadfc\uc774\ub2e4. \ub610\ud55c \\(N(W &#8216; ) = N(V)\\)\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \ud568\uc218 \\(N\\)\uc5d0 \uc758\ud558\uc5ec \\(v_j\\)\ub97c \uadfc\uc73c\ub85c \ud558\ub294 \\(W_j &#8216; \\)\uc758 \uc21c\ud658\uae30\uc800\ub294 \\(w_j\\)\ub97c \uadfc\uc73c\ub85c \ud558\ub294 \\(W_j\\)\uc758 \uc21c\ud658\uae30\uc800\uc5d0 \ub300\uc751\ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubcf4\uc870\uc815\ub9ac 8\uc744 \uc774\uc6a9\ud558\uae30 \uc704\ud574\uc11c\ub294 \\(W &#8216; = W_1 &#8216; \\oplus + \\cdots + \\oplus W_r &#8216; \\)\uc784\uc744 \ubcf4\uc5ec\uc57c \ud558\ub294\ub370, \uc774\uac83\uc744 \uc704\ud574\uc11c\ub294 \ub2e4\uc74c\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4.<br \/>\n\\[\\sum_{j=1}^r w_j &#8216; = \\mathbf{0} \\quad \\Rightarrow \\quad w_j &#8216; = \\mathbf{0} .\\tag{9}\\]<br \/>\n\uc5ec\uae30\uc11c \\(j= 1,\\,2,\\,\\cdots,\\,r\\)\uc774\uace0 \\(w_j &#8216; \\in W_j &#8216;\\)\uc774\ub2e4. \ubcf4\uc870\uc815\ub9ac 6\uc5d0 \uc758\ud558\uc5ec \\(K[t]\\)\uc758 \ub2e4\ud56d\uc2dd \\(q_1 (t) ,\\) \\(\\cdots,\\) \\(q_r (t)\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(w_j &#8216; = q_j (N) (v_j)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (8)\uc758 \ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\uc774 \ub41c\ub2e4.<br \/>\n\\[\\sum_{j=1}^r q_j (N) (v_j ) = \\mathbf{0}.\\]<br \/>\n\uc774 \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(N\\)\uc744 \ucde8\ud558\uace0 \\(N(v_j ) = w_j\\)\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\sum_{j=1}^r q_j (N) (W_j) = \\mathbf{0} .\\]<br \/>\n\uc774\ub85c\uc368 \\(N(V)\\)\ub294 \uc21c\ud658\ubd80\ubd84\uacf5\uac04 \\(W_1,\\) \\(\\cdots,\\) \\(W_r\\)\uc758 \uc9c1\ud569\uc774\uba70, \uc784\uc758\uc758 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(q_j (N)(w_j) = \\mathbf{0}\\)\uc774\ub2e4. \\(w_j\\)\uac00 \\(W_j\\)\uc758 \uc21c\ud658\uae30\uc800\uc758 \uadfc\uc774\ubbc0\ub85c \\(q_j(N)\\)\uc740 \\(W_j\\) \uc704\uc5d0\uc11c \uc601\ubcc0\ud658\uc774\ub2e4. \ubcf4\uc870\uc815\ub9ac 7\uc5d0 \uc758\ud558\uc5ec \ubaa8\ub4e0 \\(q_j (t)\\)\uac00 \\(t\\)\ub97c \uc778\uc218\ub85c \uac00\uc9c0\ubbc0\ub85c, \\(q_j (t) = q_j &#8216; (t) t\\) \uaf34\ub85c \ud45c\ud604\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\n\\sum_{j=1}^r q_j (N) (v_j)<br \/>\n&#038;= \\sum_{j=1}^r [q_j &#8216; (N)](v_j) \\\\[5pt]<br \/>\n&#038;= \\sum_{j=1}^r q_j &#8216; (N)(w_j) = \\mathbf{0}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \ub2e4\uc2dc \ubcf4\uc870\uc815\ub9ac 7\uc5d0 \uc758\ud558\uc5ec, \uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(m_j = \\dim(W_j)\\)\uc77c \ub54c \\(q_j &#8216; (t)\\)\ub294 \ub2e8\ud56d\uc2dd \\(t^{m_j}\\)\uc744 \uc778\uc218\ub85c \uac00\uc9c4\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(q_j (t)\\)\ub294 \\(t^{m_j +1}\\)\uc744 \uc778\uc218\ub85c \uac00\uc9c4\ub2e4. \uadf8\ub7ec\uba74 \\(q_j (N)\\)\uc774 \\(N^{m_j+1}\\)\uc758 \uc778\uc218\ub97c \ud3ec\ud568\ud574\uc57c \ud558\ub294\ub370, \\(N^{m_j+1}\\)\uc740 \\(V_j\\) \uc704\uc5d0\uc11c \uc601\ubcc0\ud658\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(w_j &#8216; = q_j (N)(v_j) = \\mathbf{0}\\)\uc774\ub2e4. \uc774\ub85c\uc368 \ubcf4\uc870\uc815\ub9ac 8\uc758 \uacb0\ub860\uc744 \uc5bb\uae30 \uc704\ud55c \uc870\uac74\uc774 \ubaa8\ub450 \uc131\ub9bd\ud558\ubbc0\ub85c, \\(V\\)\ub294 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc815\ub9ac 9\ub294 \uc21c\ud658\ubd80\ubd84\uacf5\uac04 \uc704\uc5d0\uc11c\uc758 \uba71\uc601\ubcc0\ud658\uc758 \ud45c\ud604\uc744 \ubca1\ud130\uacf5\uac04 \uc804\uccb4\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\uac8c \ud574\uc900\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 10.<\/span><br \/>\n\\(N\\)\uc774 \ubca1\ud130\uacf5\uac04 \\(V\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \uba71\uc601\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(B\\)\uc5d0 \ub300\ud55c \\(N\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774<br \/>\n\\[\\left(\\begin{array}{ccccc}<br \/>\n0 &#038; 1 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; \\cdots &#038; 0<br \/>\n\\end{array}\\right)\\]<br \/>\n\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \\(V\\)\uc758 \uae30\uc800 \\(B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\n\uc815\ub9ac 9\uc5d0 \uc758\ud558\uc5ec \\(V\\)\ub97c \\(N\\)-\ubd88\ubcc0\uc778 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc9c1\ud569\uc744 \uc774\ub8e8\ub294 \uac01 \uc21c\ud658\ubd80\ubd84\uacf5\uac04\uc740 \uae30\uc800\ub97c \uac16\ub294\ub370, \ud2b9\ud788 \uadf8 \uae30\uc800\uc5d0 \ub300\ud55c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \ud45c\ud604\ud589\ub82c\uc774 \uc815\ub9ac\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ud589\ub82c\uc758 \uc77c\ubd80\uc640 \uac19\ub3c4\ub85d \ud558\ub294 \uae30\uc800\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadf8\ub7ec\ud55c \ud45c\ud604\ud589\ub82c\ub4e4\uc744 \ub354\ud558\uc5ec \ub9cc\ub4e0 \ud589\ub82c\uc740 \uc815\ub9ac\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \uac83\uacfc \uac19\uc740 \uaf34\uc774 \ub41c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc815\ub9ac 4\uc640 \uc815\ub9ac 10\uc744 \uacb0\ud569\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uba4b\uc9c4 \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p class=\"marginbottom0\"><span class=\"theorem\">\uc815\ub9ac 11. (\uc870\ub974\ub2f9 \ud45c\uc900\ud615)<\/span><\/p>\n<p>\\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0 \ucc28\uc6d0\uc774 \\(n\\)\uc774\uba70 \\(T\\)\uac00 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc758 \uadfc\uc774 \\(\\lambda_1,\\) \\(\\cdots,\\) \\(\\lambda_r\\)\uc774\uace0 \uc774\ub4e4\uc774 \ubaa8\ub450 \ub2e4\ub978 \uac12\uc774\uba70 \uc774\ub4e4\uc758 \uc911\ubcf5\ub3c4\uac00 \uc21c\uc11c\ub300\ub85c \\(m_1,\\) \\(\\cdots,\\) \\(m_r\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc758 \uae30\uc800 \\(B\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774<br \/>\n\\[A = \\left(\\begin{array}{cccc}<br \/>\nA_1 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; A_2 &#038; \\cdots &#038; 0 \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\n0 &#038; 0 &#038; \\cdots &#038; A_r<br \/>\n\\end{array}\\right)\\]<br \/>\n\uc758 \uaf34\uc774 \ub41c\ub2e4. \uc5ec\uae30\uc11c \\(A_j\\)\ub294<br \/>\n\\[A_j = \\left(\\begin{array}{ccccc}<br \/>\n\\lambda_j &#038; 1 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; \\lambda_j &#038; 1 &#038; \\cdots &#038; 0 \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; \\cdots &#038; 1 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; \\cdots &#038; \\lambda_j<br \/>\n\\end{array}\\right)\\]<br \/>\n\uaf34\uc778 \\(m_j \\times m_j\\) \uc815\uc0ac\uac01\ud589\ub82c\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc704 \uc815\ub9ac\uc5d0\uc11c \\(A\\)\uc640 \uac19\uc740 \uaf34\uc744 <span class=\"defined\">\uc870\ub974\ub2f9 \ud45c\uc900\ud615<\/span>(Jordan normal form)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(A_j\\)\ub97c \uace0\uc733\uac12 \\(\\lambda_j\\)\uc5d0 \ub300\uc751\ud558\ub294 <span class=\"defined\">\uc870\ub974\ub2f9 \ube14\ub85d<\/span>(Jordan block)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc815\ub9ac 11\uc758 \uc99d\uba85.<\/span><br \/>\n\\(V\\)\ub97c \\(T\\)\uc5d0 \ub300\ud55c \ud2b9\uc131\ubd80\ubd84\uacf5\uac04 \\(U_1,\\) \\(\\cdots,\\) \\(U_r\\)\ub85c \ubd84\ud574\ud558\uc790. \uc815\ub9ac 4\uc5d0 \uc758\ud558\uc5ec \uac01 \ubd80\ubd84\uacf5\uac04 \uc704\uc5d0\uc11c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub300\uac01\ubd80\ubd84\uacfc \uba71\uc601\ubd80\ubd84\uc758 \ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \uc815\ub9ac 10\uc5d0 \uc758\ud558\uc5ec \uac01 \ud2b9\uc131\ubd80\ubd80\uacf5\uac04 \\(U_j\\)\uc758 \uae30\uc800\uac00 \uc874\uc7ac\ud558\uc5ec \uadf8 \uae30\uc800\uc5d0 \ub300\ud55c \ud589\ub82c\ud45c\ud604\uc758 \uba71\uc601\ubd80\ubd84\uc740 \ub300\uac01\uc131\ubd84\uacfc \uc778\uc811\ud55c \uc717\ucabd \uc131\ubd84\uc740 \\(1\\)\uc774\uace0 \ub2e4\ub978 \uc131\ubd84\uc740 \\(0\\)\uc778 \ud589\ub82c\uc774 \ub41c\ub2e4. \ub610\ud55c \\(U_j\\)\uc758 \ub300\uac01\ubd80\ubd84\uc740 \\(\\lambda_j I_{m_j}\\) \uaf34\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uba71\uc601\ubd80\ubd84\uacfc \ub300\uac01\ubd80\ubd84\uc744 \ub354\ud558\uba74 \\(A_j\\)\ub294 \uc815\ub9ac\uc5d0\uc11c \uc81c\uc2dc\ud55c \ud589\ub82c\uacfc \uac19\uc740 \uaf34\uc774 \ub41c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!--\n\n\n<p>Camille Jordan \uce74\ubbf8\uc720 \uc870\ub974\ub2f9<\/p>\n\n\n\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n\n\n\n<h2 class=\"margintop2\"><\/h2>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V\\) \uc704\uc5d0\uc11c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uac00 \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \\(V\\)\uc5d0 \uc5b4\ub5a0\ud55c \uae30\uc800\uac00 \uc8fc\uc5b4\uc84c\ub294\uc9c0\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4. \\(V\\)\uc640 \\( T\\)\uac00 \uc801\uc808\ud55c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(V\\)\uc758 \uae30\uc800\ub97c \uc801\uc808\ud788 \ud0dd\ud558\uc5ec \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \u2018\ub300\ub2e8\ud788 \uc88b\uc740 \ud615\ud0dc\u2019\uac00 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubca1\ud130\uacf5\uac04\uc744 \ud2b9\uc131\ubd80\ubd84\uacf5\uac04\uc758 \uc9c1\ud569\uc73c\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uacfc \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc744 \uc870\ub974\ub2f9 \ud45c\uc900\ud615\uc73c\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ubca1\ud130\uacf5\uac04\uc740 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc778 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \\( \\newcommand{\\Hom}{{\\operatorname{Hom}}} \\newcommand{\\Mat}{{\\operatorname{Mat}}} \\newcommand{\\proj}{{\\operatorname{proj}}} \\newcommand{\\adj}{{\\operatorname{adj}}} \\newcommand{\\Ker}{{\\operatorname{Ker}}} \\)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[57],"tags":[468,465,503,490,505,499,500,498,496,497,429,484,426,495,502,501,489,494,504,543,491,488],"class_list":["post-5746","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-468","tag-465","tag-503","tag-490","tag-505","tag-499","tag-500","tag-498","tag-496","tag-497","tag-429","tag-484","tag-426","tag-495","tag-502","tag-501","tag-489","tag-494","tag-504","tag-543","tag-491","tag-488"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5746","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=5746"}],"version-history":[{"count":36,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5746\/revisions"}],"predecessor-version":[{"id":6026,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5746\/revisions\/6026"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5746"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5746"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5746"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}