{"id":6601,"date":"2021-06-16T11:14:55","date_gmt":"2021-06-16T02:14:55","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=6601"},"modified":"2021-06-17T09:06:33","modified_gmt":"2021-06-17T00:06:33","slug":"linear-algebra-relation-between-multiplicities","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-relation-between-multiplicities\/","title":{"rendered":"\uace0\uc720\uacf5\uac04\uc758 \ucc28\uc6d0\uacfc \uace0\uc733\uac12\uc758 \uc911\ubcf5\ub3c4\uc758 \uad00\uacc4"},"content":{"rendered":"

\\(A\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uace0 \ubaa8\ub4e0 \uc131\ubd84\uc774 \uccb4 \\(K\\)\uc758 \uc131\ubd84\uc774\ub77c\uace0 \ud558\uc790. \uc989 \\(A\\in M_n (K)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \\(p_A (t)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\lambda \\in K\\)\uac00 \\(A\\)\uc758 \uace0\uc733\uac12\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(p_A(\\lambda)=0\\)\uc778 \uac83\uc774\ub2e4. \uc774\ub54c \\(p_A (t)\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4.
\n\\[p_A (t) = (t-\\lambda )^r q(t).\\]
\n\uc5ec\uae30\uc11c \\((t-\\lambda)\\)\uc758 \ucc28\uc218 \\(r\\)\ub97c \uace0\uc733\uac12 \\(\\lambda\\)\uc758 \ub300\uc218\uc801 \uc911\ubcf5\ub3c4<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uace0\uc733\uac12 \\(\\lambda\\)\ub97c \uace0\uc815\uc2dc\ucf1c \ub450\uace0 \ub2e4\uc74c \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790.
\n\\[\\left\\{ \\mathbf{v}\\in \\mathbb{R}^n \\,\\vert\\, A\\mathbf{v} = \\lambda \\mathbf{v} \\right\\}\\]
\n\uc774 \uc9d1\ud569\uc740 \\(\\mathbb{R}^n\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774 \ub418\ub294\ub370, \uc774 \uacf5\uac04\uc744 \uace0\uc733\uac12 \\(\\lambda\\)\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\uacf5\uac04<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uace0\uc720\uacf5\uac04\uc758 \ucc28\uc6d0\uc744 \\(\\lambda\\)\uc758 \uae30\ud558\uc801 \uc911\ubcf5\ub3c4<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ub9cc\uc57d \\(p_A(t)\\)\uac00 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc644\uc804\ud788 \uc778\uc218\ubd84\ud574\ub418\uace0, \\(A\\)\uc758 \uac01 \uace0\uc733\uac12\uc758 \ub300\uc218\uc801 \uc911\ubcf5\ub3c4\uc640 \uae30\ud558\uc801 \uc911\ubcf5\ub3c4\uac00 \uc77c\uce58\ud558\uba74 \\(A\\)\ub294 \ub300\uac01\ud654 \uac00\ub2a5\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(A\\)\uc758 \uace0\uc720\ubca1\ud130\uac00 \\(\\mathbb{R}^n\\)\uc758 \uace0\uc720\uae30\uc800\uac00 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

\uc5ec\uae30\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ubb38\uc81c\ub97c \uc0dd\uac01\ud574 \ubcf4\uc790.<\/p>\n

\n

\uc9c8\ubb38.<\/span> \\(\\lambda\\)\uac00 \\(A\\)\uc758 \uace0\uc733\uac12\uc77c \ub54c, \\(\\lambda\\)\uc758 \uae30\ud558\uc801 \uc911\ubcf5\ub3c4\ub294 \ud56d\uc0c1 \\(\\lambda\\)\uc758 \ub300\uc218\uc801 \uc911\ubcf5\ub3c4 \uc774\ud558\uc778\uac00?<\/p>\n<\/div>\n

\uc9c8\ubb38\uc5d0 \ub300\ud55c \ub2f5\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. \\(\\lambda\\)\uc758 \uace0\uc720\uacf5\uac04\uc758 \ucc28\uc6d0\uc744 \\(r\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(r=0\\)\uc774\uba74 \ub354\uc774\uc0c1 \uc99d\uba85\ud560 \uac83\uc774 \uc5c6\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(r > 0\\)\uc73c\ub85c \ub450\uace0 \uc99d\uba85\uc744 \uacc4\uc18d\ud558\uc790. \\(\\lambda\\)\uc758 \uace0\uc720\uacf5\uac04\uc758 \uae30\uc800\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \ub450\uc790.
\n\\[\\mathbf{v}_1 ,\\, \\mathbf{v}_2 ,\\, \\cdots ,\\, \\mathbf{v}_r .\\]
\n\uc774 \uae30\uc800\ub97c \ud655\uc7a5\ud558\uc5ec \\(\\mathbb{R}^n\\)\uc758 \uae30\uc800\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \ub450\uc790.
\n\\[\\mathbf{v}_1 ,\\, \\mathbf{v}_2 ,\\, \\cdots ,\\, \\mathbf{v}_r ,\\, \\mathbf{v}_{r+1} ,\\, \\cdots ,\\, \\mathbf{v}_n .\\]
\n\uc774\uc81c \\(\\mathbf{v}_j\\)\ub4e4\uc744 \uc5f4\ubca1\ud130\ub85c \uac16\ub294 \ud589\ub82c\uc744 \\(P\\)\ub85c \ub450\uc790. \uc989
\n\\[P = \\left[ \\mathbf{v}_1 \\,\\, \\mathbf{v}_2 \\,\\, \\cdots \\,\\,\\mathbf{v}_r \\,\\, \\mathbf{v}_{r+1} \\,\\, \\cdots \\,\\, \\mathbf{v}_n \\right].\\]
\n\uadf8\ub7ec\uba74 \ud589\ub82c\uacf1 \\(AP\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ub41c\ub2e4.
\n\\[AP = \\left[ \\lambda\\mathbf{v}_1 \\,\\, \\lambda \\mathbf{v}_2 \\,\\, \\cdots \\,\\, \\lambda \\mathbf{v}_r \\,\\, \\ast \\,\\, \\cdots \\,\\, \\ast \\right].\\]
\n\uc704 \uc2dd\uc5d0\uc11c \\(\\ast\\)\ub294 \\(r+1\\)\uc5f4\uacfc \uadf8 \uc624\ub978\ucabd\uc758 \uc5f4\ubca1\ud130\ub97c \ub098\ud0c0\ub0b8\ub2e4. \uc774\uc81c
\n\\[B = P^{-1}AP\\]
\n\ub77c\uace0 \ud558\uc790. [\\(P\\)\ub294 \uc5f4\ubca1\ud130\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\ubbc0\ub85c \uac00\uc5ed\uc774\ub2e4.] \uadf8\ub7ec\uba74 \\(B\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ubaa8\uc591\uc774\ub2e4.
\n\\[B = P^{-1}AP =\\left[ \\lambda \\mathbf{e}_1 \\,\\, \\lambda \\mathbf{e}_2 \\,\\, \\cdots \\,\\, \\lambda \\mathbf{e}_r \\,\\, \\ast \\,\\,\\cdots \\,\\, \\ast \\right].
\n\\]
\n\uc989 \\(B\\)\uc758 \uc67c\ucabd \uc704\uc5d0 \ud06c\uae30\uac00 \\(r\\times r\\)\uc774\uace0 \ubaa8\ub4e0 \ub300\uac01\uc131\ubd84\uc774 \\(\\lambda\\)\uc774\uba70 \ub2e4\ub978 \uc131\ubd84\uc740 \\(0\\)\uc778 \uc815\uc0ac\uac01 \ube14\ub85d\ud589\ub82c\uc774 \uc788\uace0, \uc774 \ube14\ub85d\ud589\ub82c \uc544\ub798\ucabd \uc131\ubd84\uc740 \ubaa8\ub450 \\(0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(B\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd \\(p_B (t)\\)\ub294 \uc778\uc218 \\((t-\\lambda)\\)\ub97c \\(r\\)\uac1c \uc774\uc0c1 \uac00\uc9c4\ub2e4. \uc989 \\(\\lambda\\)\ub294 \\(B\\)\uc758 \uace0\uc733\uac12\uc774\uba70 \uadf8 \ub300\uc218\uc801 \uc911\ubcf5\ub3c4\ub294 \\(r\\) \uc774\uc0c1\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(A\\)\uc640 \\(B\\)\ub294 \ub2ee\uc740 \ud589\ub82c\uc774\ubbc0\ub85c \ub450 \ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A\\)\uc758 \uace0\uc733\uac12 \\(\\lambda\\) \ub610\ud55c \ub300\uc218\uc801 \uc911\ubcf5\ub3c4\uac00 \\(r\\) \uc774\uc0c1\uc774\ub2e4.<\/p>\n","protected":false},"excerpt":{"rendered":"

\\(A\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uace0 \ubaa8\ub4e0 \uc131\ubd84\uc774 \uccb4 \\(K\\)\uc758 \uc131\ubd84\uc774\ub77c\uace0 \ud558\uc790. \uc989 \\(A\\in M_n (K)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \\(p_A (t)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\lambda \\in K\\)\uac00 \\(A\\)\uc758 \uace0\uc733\uac12\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(p_A(\\lambda)=0\\)\uc778 \uac83\uc774\ub2e4. \uc774\ub54c \\(p_A (t)\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4. \\(p_A (t) = (t-\\lambda )^r q(t).\\) \uc5ec\uae30\uc11c \\((t-\\lambda)\\)\uc758 \ucc28\uc218 \\(r\\)\ub97c \uace0\uc733\uac12 \\(\\lambda\\)\uc758 \ub300\uc218\uc801 \uc911\ubcf5\ub3c4\ub77c\uace0 \ubd80\ub978\ub2e4. \uace0\uc733\uac12 \\(\\lambda\\)\ub97c \uace0\uc815\uc2dc\ucf1c \ub450\uace0 \ub2e4\uc74c \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790. \\(\\left\\{ \\mathbf{v}\\in \\mathbb{R}^n \\,\\vert\\, A\\mathbf{v} =…<\/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":[557,420,418,419],"class_list":["post-6601","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-diagonalization","tag-eigenspace","tag-eigenvalue","tag-eigenvector"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6601","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=6601"}],"version-history":[{"count":11,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6601\/revisions"}],"predecessor-version":[{"id":6618,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6601\/revisions\/6618"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6601"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=6601"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=6601"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}