{"id":5453,"date":"2020-10-29T14:23:04","date_gmt":"2020-10-29T05:23:04","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5453"},"modified":"2020-12-20T20:55:32","modified_gmt":"2020-12-20T11:55:32","slug":"linear-algebra-eigen-values-and-eigen-vectors","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-eigen-values-and-eigen-vectors\/","title":{"rendered":"\uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130"},"content":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\uc758 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf4\uace0 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \uace0\uc733\uac12\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubd05\ub2c8\ub2e4. \ub610\ud55c \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uacfc \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc758 \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \uc2a4\ud399\ud2b8\ub7fc \ubd84\ud574 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubd05\ub2c8\ub2e4.<\/p>\n

\n\\[
\n\\newcommand{\\parallelsym}{\\mathbin{\\!\/\\mkern-5mu\/\\!}}
\n\\]\n<\/div>\n

<\/p>\n

\uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\uc758 \ub73b<\/h2>\n

\\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T : V \\rightarrow V\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc2a4\uce7c\ub77c \\(\\lambda \\in K\\)\uc640 \uc601\ubca1\ud130\uac00 \uc544\ub2cc \ubca1\ud130 \\(v\\in V\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[T(v) = \\lambda v\\tag{1}\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(\\lambda\\)\ub97c \\(T\\)\uc758 \uace0\uc733\uac12<\/span>(eigenvalue)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \uc774\uc640 \uac19\uc740 \\(\\lambda\\)\uc5d0 \ub300\ud558\uc5ec \ub4f1\uc2dd (1)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294, \uc601\ubca1\ud130\uac00 \uc544\ub2cc \ubca1\ud130 \\(v\\)\ub97c \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130<\/span>(eigenvector)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\u2018eigen\u2019\uc740 \ub3c5\uc77c\uc5b4\uc774\uba70 \u2018\uc544\uc774\uac90\u2019\uc774\ub77c\uace0 \uc77d\ub294\ub2e4. \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\ub97c \uac01\uac01 \ud2b9\uc131\uac12(characteristic value), \ud2b9\uc131\ubca1\ud130(characteristic vector)\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

\uae30\ud558\ud559\uc801\uc73c\ub85c \ubcf4\uba74 \\(T\\)\uc758 \uace0\uc720\ubca1\ud130\ub294 \\(T(v) \\parallelsym v\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubca1\ud130 \\(v\\)\uc774\ub2e4.<\/p>\n

\n

\ubcf4\uae30 1.<\/span>
\n\\(V = \\mathbb{R}^2\\)\ub77c\uace0 \ud558\uace0
\n\\[A = \\left(
\n\\begin{matrix}
\n0 & 1 \\\\ 1 & 0
\n\\end{matrix}
\n\\right)\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc120\ud615\ubcc0\ud658 \\(T_A : V \\rightarrow V\\)\ub97c \ud589\ub82c \\(A\\)\ub97c \uc67c\ucabd\uc5d0 \uacf1\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\begin{align}
\nT \\left(\\matrix{ +1 \\\\ +1}\\right) &= +1 \\cdot \\left(\\matrix{+1\\\\+1}\\right), \\\\[5pt]
\nT \\left(\\matrix{ +1 \\\\ -1}\\right) &= -1 \\cdot \\left(\\matrix{+1\\\\-1}\\right)
\n\\end{align}\\]
\n\uc774\ubbc0\ub85c \\(+1\\)\uacfc \\(-1\\)\uc740 \ubaa8\ub450 \\(T_A\\)\uc758 \uace0\uc733\uac12\uc774\uba70, \uac01 \uace0\uc733\uac12\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub294 \uc704 \uc2dd\uacfc \uac19\ub2e4.\n<\/p>\n<\/div>\n

\uc704 \ubcf4\uae30\uc640 \uac19\uc774 \ud589\ub82c \\(A\\)\ub97c \uc67c\ucabd\uc5d0 \uacf1\ud558\ub294 \uc120\ud615\ubcc0\ud658 \\(T_A\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \u2018\\(T_A\\)\uc758 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\u2019\ub97c \uac04\ub2e8\ud558\uac8c \u2018\\(A\\)\uc758 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\u2019\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n

\n

\ubcf4\uae30 2.<\/span>
\n\uc2e4\ubca1\ud130\uacf5\uac04 \\(V=C^\\infty (\\mathbb{R})\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubbf8\ubd84\uc5f0\uc0b0\uc790 \\(D\\)\ub97c \uc0dd\uac01\ud558\uc790. \\(D\\)\ub294 \\(V\\) \uc704\uc5d0\uc11c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uc784\uc758\uc758 \\(\\lambda\\in\\mathbb{R}\\)\ub294 \ubbf8\ubd84\uc5f0\uc0b0\uc790\uc758 \uace0\uc733\uac12\uc774\uba70, \uadf8\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub294 \\(x\\mapsto Ce^{\\lambda x}\\) (\\(C\\)\ub294 \uc784\uc758\uc758 \uc0c1\uc218)\uc774\ub2e4.<\/p>\n<\/div>\n

\uc77c\ubc18\uc801\uc73c\ub85c \uace0\uc815\ub41c \uace0\uc733\uac12 \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub4e4\uc758 \ubaa8\uc784\uc5d0 \uc601\ubca1\ud130\ub97c \ucd94\uac00\ud55c \uc9d1\ud569\uc740 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774 \ub41c\ub2e4. \uc774 \uacf5\uac04\uc744 \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\uacf5\uac04<\/span>(eigenspace)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(T\\)\uc758 \ud575(kernel)\uc758 \ucc28\uc6d0\uc774 \\(1\\) \uc774\uc0c1\uc774\uba74 \\(\\lambda = 0\\)\uc740 \\(T\\)\uc758 \uace0\uc733\uac12\uc774 \ub418\ub294\ub370, \uc774\ub54c \\(0\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\uacf5\uac04\uc740 \\(T\\)\uc758 \ud575\uacfc \uac19\ub2e4. \uc774\ub7ec\ud55c \uc758\ubbf8\uc5d0\uc11c \ud575\uc740 \uace0\uc720\uacf5\uac04\uc758 \ud2b9\uc218\ud55c \uacbd\uc6b0\ub85c\uc11c \ub2e4\ub8e8\uc5b4\uc9c8 \uc218 \uc788\ub2e4.<\/p>\n

\ub9cc\uc57d \\(T\\)\uc758 \uace0\uc720\ubca1\ud130\ub4e4\uc744 \uc774\uc6a9\ud558\uc5ec \\(V\\)\uc758 \uae30\uc800\ub97c \uad6c\uc131\ud560 \uc218 \uc788\uc73c\uba74, \uadf8\ub807\uac8c \uad6c\uc131\ub41c \uae30\uc800\ub97c \\(T\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 \uace0\uc720\uae30\uc800<\/span>(eigenbasis)\ub77c\uace0 \ubd80\ub978\ub2e4. (\uace0\uc720\uae30\uc800\uac00 \ud56d\uc0c1 \uc874\uc7ac\ud558\uc9c0\ub294 \uc54a\ub294\ub2e4.) \uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c\uacfc \uace0\uc720\uae30\uc800, \uace0\uc733\uac12\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uad00\uacc4\uac00 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1.<\/span>
\n\\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T:V\\rightarrow V\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(T\\)\ub97c \ub300\uac01\ud589\ub82c\ub85c \ud45c\ud604\ud560 \uc218 \uc788\uc744 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(T\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 \uace0\uc720\uae30\uc800\uac00 \uc874\uc7ac\ud558\ub294 \uac83\uc774\ub2e4. \uc774\ub54c \\(T\\)\ub97c \ud45c\ud604\ud558\ub294 \ub300\uac01\ud589\ub82c\uc758 \uc131\ubd84\ub4e4\uc774 \\(T\\)\uc758 \uace0\uc733\uac12\uc774\ub2e4. [\uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c, \uc8fc\uc5b4\uc9c4 \uc120\ud615\ubcc0\ud658\uc5d0 \ub300\ud55c \uace0\uc720\uae30\uc800\uac00 \uc874\uc7ac\ud560 \ub54c \uadf8 \uc120\ud615\ubcc0\ud658\uc744 \ub300\uac01\ud654 \uac00\ub2a5<\/span>(diagonalizable)\ud55c \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.]\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(v_1 ,\\) \\(v_2 ,\\) \\(\\cdots ,\\) \\(v_n\\)\uc774 \uace0\uc720\uae30\uc800 \\(B\\)\uc758 \uc6d0\uc18c\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc774 \ubca1\ud130\ub4e4\uc5d0 \ub530\ub77c\ub294 \uace0\uc733\uac12\ub4e4\uc744 \\(\\lambda_1 ,\\) \\(\\lambda_2 ,\\) \\(\\cdots ,\\) \\(\\lambda_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[T(v_j) = \\lambda_j v_j\\tag{2}\\]
\n\uc774\uba70, \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub300\uac01\uc131\ubd84\uc774 \\(\\lambda_1,\\) \\(\\lambda_2 ,\\) \\(\\cdots ,\\) \\(\\lambda_n\\)\uc778 \ub300\uac01\ud589\ub82c\uc774\ub2e4. \uc5ed\uc73c\ub85c, \ub9cc\uc57d \uc8fc\uc5b4\uc9c4 \uae30\uc800\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \ub300\uac01\ud589\ub82c\uc774\uba74, \uc8fc\uc5b4\uc9c4 \uac01 \uae30\uc800\uc6d0\uc18c \\(v_i\\)\uc5d0 \ub300\ud558\uc5ec \uc801\ub2f9\ud55c \uc2a4\uce7c\ub77c \\(\\lambda_i\\)\uac00 \uc874\uc7ac\ud558\uc5ec (2)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c, \uc8fc\uc5b4\uc9c4 \uae30\uc800\ub294 \uace0\uc720\uae30\uc800\uc774\ub2e4.
\n<\/span>\n<\/p>\n<\/div>\n

\ub9cc\uc57d \\(B\\)\uc640 \\(B ‘ \\)\uc774 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \uae30\uc800\uc774\uba74, \\(V\\)\uc758 \uc784\uc758\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[M_{B ‘} (T) = P^{-1} M_B (T) P\\tag{3}\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(P\\)\ub294 \\(B ‘ \\)\uc73c\ub85c\ubd80\ud130 \\(B\\)\ub85c\uc758 \uae30\uc800\ubcc0\ud658\ud589\ub82c(transition matrix)\uc774\ub2e4. \uc774 \uc2dd\uacfc \uac19\uc774 \ub450 \ud589\ub82c \\(M ,\\) \\(M ‘ \\)\uc5d0 \ub300\ud558\uc5ec \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(M = P^{-1} M ‘ P\\)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc744 \ub54c, \uc774\ub4e4 \ub450 \ud589\ub82c\uc744 \uc11c\ub85c \ub2ee\uc74c\ud589\ub82c<\/span>(similar)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \ud589\ub82c \\(M\\)\uc744 \uc120\ud615\ubcc0\ud658\uc73c\ub85c \ubcf4\uc558\uc744 \ub54c, \ubca1\ud130\uacf5\uac04\uc758 \uae30\uc800\ub97c \ubc14\uafc8\uc73c\ub85c\uc368 \ub3d9\uc77c\ud55c \uc120\ud615\ubcc0\ud658\uc5d0 \ub300\ud55c \ub2e4\ub978 \ud589\ub82c \\(M ‘ \\)\uc744 \uc5bb\uc744 \uc218 \uc788\uc744 \ub54c, \ub450 \ud589\ub82c \\(M,\\) \\(M ‘ \\)\uc740 \ub2ee\uc74c\ud589\ub82c\uc774\ub2e4.<\/p>\n

\uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc815\ub9ac 1\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

\n

\ub530\ub984\uc815\ub9ac 2.<\/span>
\n\\(A\\in M_n (K)\\)\uac00 \ub300\uac01\ud589\ub82c\uacfc \ub2ee\uc74c\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A\\)\uc5d0 \ub300\ud55c \\(K^n\\)\uc758 \uace0\uc720\uae30\uc800\uac00 \uc874\uc7ac\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 3.<\/span>
\n\uc55e\uc758 \ubcf4\uae30 1\uc5d0\uc11c \uad6c\ud55c \ub450 \uace0\uc720\ubca1\ud130 \\((1,\\,1),\\) \\((1,\\,-1)\\)\uc740 \\(\\mathbb{R}^2\\)\uc758 \uae30\uc800\ub97c \uc774\ub8e8\ubbc0\ub85c, \uc774\ub4e4\uc740 \\(\\mathbb{R}^2\\)\uc758 \uace0\uc720\uae30\uc800\uc774\ub2e4. \ub450 \ubca1\ud130\ub294 \uc11c\ub85c \uc218\uc9c1\uc778\ub370, \uc774\uac83\uc740 \uc6b0\uc5f0\uc774 \uc544\ub2c8\ub2e4. \uc774\uc5d0 \ub300\ud55c \uc790\uc138\ud55c \ub0b4\uc6a9\uc740 \ub4a4\uc5d0\uc11c \uc0b4\ud3b4\ubcfc \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 4.<\/span>
\n\ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658 \\(T_\\theta : \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2\\)\ub97c \uc0b4\ud3b4\ubcf4\uc790.
\n\\[T_\\theta = \\left(
\n\\begin{array}{cr} \\cos\\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{array}
\n\\right).\\tag{4}\\]
\n\uc774 \ubcc0\ud658\uc740 \ud3c9\uba74\uc758 \uc810\uc744 \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \\(\\theta\\)\ub9cc\ud07c \ud68c\uc804\uc2dc\ud0a4\ub294 \ud68c\uc804\uc2dc\ud0a4\ub294 \ubcc0\ud658\uc774\ub2e4. \\(\\theta\\)\uac00 \\(\\pi\\)\uc758 \uc815\uc218\ubc30\uac00 \uc544\ub2cc \uacbd\uc6b0\ub97c \uc0dd\uac01\ud574 \ubcf4\uc790. \uc774 \uacbd\uc6b0 \uc601\ubca1\ud130\uac00 \uc544\ub2cc \ubca1\ud130\ub97c \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uc790\uae30 \uc790\uc2e0\uacfc \ud3c9\ud589\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(T_\\theta\\)\ub294 \uace0\uc733\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \uacbd\uc6b0\uc5d0\ub294 \uace0\uc720\uae30\uc800\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\uba70, \\(T_\\theta\\)\ub294 \ub300\uac01\ud654 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4.\n<\/p>\n<\/div>\n

<\/p>\n

\ud2b9\uc131\ub2e4\ud56d\uc2dd<\/h2>\n

\uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\ub97c \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

\n

\uc815\uc758.<\/span>
\n\\(A\\in M_n (K)\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(n\\)\ucc28 \ub2e4\ud56d\uc2dd
\n\\[p(t) = \\det (tI_n – A)\\]
\n\ub97c \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd<\/span>(characteristic polynomial)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n

\\(A\\in M_n (K)\\)\uc774\uace0 \\(p(t)\\)\uac00 \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(A\\)\uc758 \uace0\uc733\uac12\uc744 \uad6c\ud558\uae30 \uc704\ud574\uc11c\ub294 \ubc29\uc815\uc2dd
\n\\[p(t)=0\\tag{5}\\]
\n\uc758 \uadfc\uc744 \uad6c\ud558\uba74 \ub41c\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\lambda \\in K\\)\uc77c \ub54c, \\(\\lambda\\)\uac00 \\(p(t)=0\\)\uc758 \uadfc\uc778 \uac83\uacfc \\(\\lambda\\)\uac00 \\(A\\)\uc758 \uace0\uc733\uac12\uc778 \uac83 \uc0ac\uc774\uc5d0 \ub2e4\uc74c\uacfc \uac19\uc740 \uad00\uacc4\uac00 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.
\n\\[\\begin{align}
\np(\\lambda) = 0
\n& \\,\\, \\Leftrightarrow \\,\\,
\n\\det(\\lambda I_n – A) = 0 \\\\[5pt]
\n& \\,\\, \\Leftrightarrow \\,\\,
\n(\\lambda I_n – A) \\text{ is singular (non-invertible)} \\\\[5pt]
\n& \\,\\, \\Leftrightarrow \\,\\,
\n\\exists \\mathbf{x} \\in K^n : \\, (\\mathbf{x} \\ne \\mathbf{0} \\text{ and } (\\lambda I_n -A) = \\mathbf{0}) \\\\[5pt]
\n& \\,\\, \\Leftrightarrow \\,\\,
\n\\exists \\mathbf{x} \\in K^n : \\, (\\mathbf{x} \\ne \\mathbf{0} \\text{ and } A\\mathbf{x} = \\lambda\\mathbf{x}) .\\tag{6}
\n\\end{align}\\]
\n\ub354\uc6b1\uc774 \uc774 \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130 \uace0\uc733\uac12 \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\uacf5\uac04\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc5bb\ub294\ub2e4. \uc989, \uace0\uc733\uac12 \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\uacf5\uac04\uc740 \ub3d9\ucc28\ubc29\uc815\uc2dd
\n\\[(\\lambda I_n – A)\\mathbf{x} = \\mathbf{0}\\tag{7}\\]
\n\uc758 \ud574\uacf5\uac04\uc744 \uad6c\ud558\ub294 \uac83\uacfc \uac19\ub2e4.<\/p>\n

\uc774\ub7ec\ud55c \uacb0\uacfc\ub294 \uccb4\uac00 \ubb34\uc5c7\uc778\uc9c0\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4. \uc65c\ub0d0\ud558\uba74, \uc608\ucee8\ub300 \ubc29\uc815\uc2dd \\(t^2 + 1 = 0\\)\uc740 \uc2e4\uc218\uccb4 \\(\\mathbb{R}\\)\uc5d0\uc11c\ub294 \ud574\ub97c \uac16\uc9c0 \uc54a\uc9c0\ub9cc \ubcf5\uc18c\uc218\uccb4 \\(\\mathbb{C}\\)\uc5d0\uc11c\ub294 \ud574\ub97c \uac16\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(t^2 + 1\\)\uc774\uba74 \uccb4\uac00 \ubb34\uc5c7\uc778\uc9c0\uc5d0 \ub530\ub77c \uace0\uc733\uac12\uc774 \uc874\uc7ac\ud560 \uc218\ub3c4 \uc788\uace0 \uadf8\ub807\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n

\n

\ubcf4\uae30 5.<\/span>
\n\ubcf4\uae30 1\uc5d0\uc11c \ubcf8 \ud589\ub82c\uc744 \ub2e4\uc2dc \uc0b4\ud3b4\ubcf4\uc790.
\n\\[A = \\left(
\n\\begin{matrix}
\n0 & 1 \\\\ 1 & 0
\n\\end{matrix}
\n\\right)\\]
\n\\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[p(t) = \\det(tI_2 – A) = \\det
\n\\left(\\begin{array}{rr}
\nt & -1 \\\\ -1 & t
\n\\end{array}\\right)
\n=t^2 -1.\\]
\n\ubc29\uc815\uc2dd \\(p(t)=0\\)\uc758 \uadfc\uc740 \\(+1,\\) \\(-1\\)\uc774\uba70 \uc774 \ub450 \uac12\uc774 \uace7 \\(A\\)\uc758 \uace0\uc733\uac12\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 6.<\/span>
\n\ub2e4\uc74c \ud589\ub82c\uc744 \uc0b4\ud3b4\ubcf4\uc790.
\n\\[B = \\left(\\begin{array}{cr}
\n0 & -1 \\\\ 1 & 0
\n\\end{array}\\right).\\]
\n\uc774 \ud589\ub82c\uc740 \\(\\pi\/2\\)\ub9cc\ud07c \ud68c\uc804\ud558\ub294 \ubcc0\ud658\uc774\ub2e4. \uc774 \ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740
\n\\[p(t) = t^2 + 1\\]
\n\uc778\ub370, \ubc29\uc815\uc2dd \\(p(t)=0\\)\uc740 \uc2e4\uc218 \ubc94\uc704\uc5d0\uc11c \uadfc\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(B\\)\ub294 \uc2e4\uc218\uccb4 \uc704\uc5d0\uc11c \uace0\uc733\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 7.<\/span>
\n\ub2e4\uc74c \uac19\uc740 \ud589\ub82c\uc744 \uc0b4\ud3b4\ubcf4\uc790.
\n\\[C = \\left(\\begin{array}{ccc}
\n5 & 1 & 2 \\\\ 1 & 6 & 1 \\\\ 2 & 3 & 7
\n\\end{array}\\right).\\]
\n\\(C\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \uad6c\ud558\uba74
\n\\[p(t) = (-4+t)(43 – 14t + t^2 )\\]
\n\uc774\ubbc0\ub85c, \\(4\\)\ub294 \\(A\\)\uc758 \uace0\uc733\uac12\uc774\ub2e4. \\(4\\)\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\uacf5\uac04\uc744 \uad6c\ud558\ub824\uba74 \ub3d9\ucc28\ubc29\uc815\uc2dd
\n\\[(4I_3 – A)\\mathbf{x} = \\mathbf{0}\\]
\n\uc758 \ud574\ub97c \uad6c\ud574\uc57c \ud55c\ub2e4. \uc774 \ubc29\uc815\uc2dd\uacfc \ub3d9\uce58\uc778 \ub2e4\uc74c \ubc29\uc815\uc2dd\uc744 \ud480\uc790.
\n\\[(A-4I_3 )\\mathbf{x} = \\mathbf{0}\\]
\n\uc774 \ubc29\uc815\uc2dd\uc744 \ud589\ub82c\ub85c \ud45c\ud604\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\left(\\begin{array}{} 1 & 1 & 2 \\\\ 1 & 2 & 1 \\\\ 2 & 3 & 3 \\end{array}\\right)
\n\\left(\\begin{array}{} x_1 \\\\ x_2 \\\\ x_3 \\end{array}\\right) =
\n\\left(\\begin{array}{} 0 \\\\ 0 \\\\ 0 \\end{array}\\right).\\]
\n\uc774 \ubc29\uc815\uc2dd\uc758 \ud574\uacf5\uac04\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\left\\{ (-3z,\\,z,\\,z) \\,\\vert\\, z\\in\\mathbb{R}\\right\\}\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(4\\)\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\uacf5\uac04\uc740 \ud558\ub098\uc758 \ubca1\ud130 \\((-3,\\,1,\\,1)\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \\(1\\)\ucc28\uc6d0 \uacf5\uac04\uc774\ub2e4.\n<\/p>\n<\/div>\n

\\(T\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \\(V\\)\uc5d0 \uae30\uc800\uac00 \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c, \uadf8 \uae30\uc800\ub97c \uae30\uc900\uc73c\ub85c \\(T\\)\ub97c \ud589\ub82c \\(A\\)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \\(A\\)\ub294 \\(V\\)\uc758 \uae30\uc800\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c0\uc9c0\ub9cc \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \uae30\uc800\uc640\ub294 \ub3c5\ub9bd\uc801\uc774\ub2e4.<\/p>\n

\n

\uc815\ub9ac 2.<\/span>
\n\ub450 \ud589\ub82c\uc774 \uc11c\ub85c \ub2ee\uc740 \ud589\ub82c\uc774\uba74, \ub450 \ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \uc77c\uce58\ud55c\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(A\\)\uc640 \\(B\\)\uac00 \uc11c\ub85c \ub2ee\uc740 \ud589\ub82c\uc774\uace0 \\(B = P^{-1} AP\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud589\ub82c \uacc4\uc0b0\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\ntI_n – B
\n&= tI_n – P^{-1}AP \\\\[5pt]
\n&= P^{-1} (tI_n) P – P^{-1} AP \\\\[5pt]
\n&= P^{-1} (tI_n -A)P.
\n\\end{align}\\]
\n\ud589\ub82c\uc2dd\uc740 \uacf1\uc744 \ubcf4\uc874\ud558\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\det(tI_n – B)
\n&= \\det(P^{-1})\\det(tI_n – A)\\det(P) \\\\[5pt]
\n&= \\det(P)^{-1} \\det(tI_n – A)\\det(P) \\\\[5pt]
\n&= \\det(tI_n -A).
\n\\end{align}\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uacfc \\(B\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \uc77c\uce58\ud55c\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\uadf8\ub7ec\ubbc0\ub85c \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T:V \\rightarrow V\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd<\/span>\uc774\ub77c \ud568\uc740 \\(T\\)\ub97c \ub098\ud0c0\ub0b4\ub294 \ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \uc774\ub974\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 3.<\/span>
\n\\(\\lambda_1,\\) \\(\\lambda_2,\\) \\(\\cdots,\\) \\(\\lambda_r\\)\uac00 \\(T : V \\rightarrow V\\)\uc758 \uace0\uc733\uac12\uc774\uace0 \ubaa8\ub450 \uc11c\ub85c \ub2e4\ub978 \uac12\uc774\uba70, \uc774\ub4e4 \uace0\uc733\uac12\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub4e4\uc744 \uc21c\uc11c\ub300\ub85c \\(v_1 ,\\) \\(v_2 ,\\) \\(\\cdots ,\\) \\(v_r\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(v_1 ,\\) \\(v_2,\\) \\(\\cdots,\\) \\(v_r\\)\ub294 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\uacb0\ub860\uacfc \ubc18\ub300\ub85c \\(v_1 ,\\) \\(v_2 ,\\) \\(\\cdots ,\\) \\(v_r\\)\uac00 \uc77c\ucc28\uc885\uc18d\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \uc774\ub4e4 \ubca1\ud130\ub4e4\uc758 \uc885\uc18d \uad00\uacc4 \ud45c\ud604 \uc911 \uac00\uc7a5 \uc9e7\uc740 \ud45c\ud604\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b4\uc790.
\n\\[v_1 = \\mu_2 v_2 + \\cdots + \\mu_s v_s .\\tag{8}\\]
\n\ub2e8, \uc774 \uc2dd\uc5d0\uc11c \ubaa8\ub4e0 \\(\\mu_j \\in K\\)\ub294 \\(0\\)\uc774 \uc544\ub2cc \uc2a4\uce7c\ub77c\uc774\ub2e4. \uc2dd\uc758 \uc591\ubcc0\uc744 \\(T\\)\uc5d0 \ub300\uc785\ud558\uc5ec \uac12\uc744 \uad6c\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\lambda_1 v_1 = \\mu_2 \\lambda_2 v_2 + \\cdots + \\mu_s \\lambda_s v_s .\\tag{9}\\]
\n(9)\ub85c\ubd80\ud130 (8)\uc5d0 \\(\\lambda_1\\)\uc744 \uacf1\ud55c \uc2dd\uc744 \ubcc0\ub9c8\ub2e4 \ube7c\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\mathbf{0} = \\mu_2 (\\lambda_2 – \\lambda_1) v_2 + \\cdots + \\mu_s (\\lambda_s – \\lambda_1 )v_s .\\tag{10}\\]
\n\uadf8\ub7f0\ub370 \\(\\lambda_j\\)\ub4e4\uc774 \ubaa8\ub450 \uc11c\ub85c \ub2e4\ub974\uace0 \\(\\mu_j\\)\uac00 \\(0\\)\uc774 \uc544\ub2c8\ubbc0\ub85c (10)\uc740 \\(v_j\\)\ub4e4\uc758 \uc885\uc18d \uad00\uacc4 \ud45c\ud604 \uc911 (8)\ubcf4\ub2e4 \ub354 \uc9e7\uc740 \ud45c\ud604\uc774\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(v_1 ,\\) \\(v_2 ,\\) \\(\\cdots ,\\) \\(v_r\\)\ub294 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\uc815\ub9ac 4.<\/span>
\n\\(A\\in M_n(K)\\)\uc774\uace0 \\(p(t)\\)\uac00 \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \ubc29\uc815\uc2dd \\(p(t)=0\\)\uc774 \uccb4 \\(K\\)\uc5d0\uc11c \\(n\\)\uac1c\uc758 \uc11c\ub85c \ub2e4\ub978 \uadfc\uc744 \uac00\uc9c0\uba74 \\(A\\)\ub294 \ub300\uac01\ud654 \uac00\ub2a5\ud558\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\uc815\ub9ac\uc758 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(A\\)\ub294 \\(n\\)\uac1c\uc758 \uc11c\ub85c \ub2e4\ub978 \uace0\uc733\uac12\uc744 \uac00\uc9c0\uba70, \uadf8 \uace0\uc733\uac12\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub294 \\(n\\)\uac1c\uc758 \uc77c\ucc28\ub3c5\ub9bd\uc778 \ubca1\ud130\uc774\ub2e4. \uc774\ub4e4 \ubca1\ud130\uac00 \uc0dd\uc131\ud558\ub294 \uacf5\uac04\uc740 \\(n\\)\ucc28\uc6d0\uc774\ubbc0\ub85c, \uc774 \uace0\uc720\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc758 \uae30\uc800\uac00 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A\\)\ub294 \ub300\uac01\ud654 \uac00\ub2a5\ud558\ub2e4.
\n<\/span><\/p>\n<\/div>\n

<\/p>\n

\uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uacfc \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658<\/h2>\n

\uc9c0\uae08\ubd80\ud130 \ub0b4\uc801\uacf5\uac04\uc774\ub77c \ud568\uc740 \uc2e4\ub0b4\uc801\uc774\ub098 \ubcf5\uc18c\ub0b4\uc801\uc774 \uc8fc\uc5b4\uc9c4 \ubca1\ud130\uacf5\uac04\uc744 \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud558\uc790.<\/p>\n

\\(V\\)\uac00 \ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(T : V \\rightarrow V\\)\uac00 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(V\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T^*\\)\uac00 \uc870\uac74
\n\\[\\langle T(u) \\,\\vert\\, v \\rangle = \\langle u \\,\\vert\\, T^* (v) \\rangle \\quad\\text{for all } u,\\,v\\in V\\]
\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(T^*\\)\ub97c \\(T\\)\uc758 \uc218\ubc18\ubcc0\ud658<\/span>(adjoint)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc784\uc758\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uc5d0 \ub300\ud558\uc5ec, \\(T\\)\uc758 \uc218\ubc18\ubcc0\ud658 \\(T^*\\)\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4. \ub610\ud55c \uc218\ubc18\ubcc0\ud658\uc758 \uc218\ubc18\ubcc0\ud658\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

\n

\uc815\ub9ac 5.<\/span>
\n\\(T^{**} = T.\\)\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(u,\\) \\(v\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc9c4 \ubca1\ud130\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\langle T(u) \\,\\vert\\,v \\rangle = \\langle u\\,\\vert\\, T^* (v) \\rangle = \\overline{\\langle T^*(v) \\,\\vert\\, u \\rangle} = \\overline{\\langle v\\,\\vert\\,T^{**}(u)\\rangle} = \\langle T^{**} (u) \\,\\vert\\, v\\rangle.\\]
\n\uc989 \uc784\uc758\uc758 \ubca1\ud130 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(T(u)\\)\uc640 \\(T^{**}(u)\\)\uac00 \uac19\uc740 \ub0b4\uc801\uac12\uc744 \uac00\uc9c0\ubbc0\ub85c \\(T(u) – T^{**}(u)\\)\ub294 \\(V\\)\uc758 \ubaa8\ub4e0 \ubca1\ud130\uc640 \uc218\uc9c1\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(T(u) – T^{**}(u) = \\mathbf{0}\\)\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\\(A = (a_{ij})\\)\uac00 \ubcf5\uc18c\ud589\ub82c\uc77c \ub54c \uadf8 \ucf24\ub808\ud589\ub82c<\/span>(conjugate)\ub97c
\n\\[\\overline{A} = \\left( \\overline{a_{ij}} \\right)\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\ucf24\ub808\ud589\ub82c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \ud569\uacfc \uacf1\uc774 \uc815\uc758\ub420 \ub54c, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\overline{A+B} = \\overline{A} + \\overline{B} ,\\,\\, \\overline{AB} = \\overline{A} \\cdot \\overline{B}.\\]
\n\\(A = (a_{ij})\\)\uac00 \ubcf5\uc18c\ud589\ub82c\uc77c \ub54c \uadf8 \ucf24\ub808\uc804\uce58\ud589\ub82c<\/span>(conjugate transpose)\ub97c
\n\\[A^* = \\,^t (\\overline{A})\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \\(A\\)\uac00 \uc2e4\ud589\ub82c\uc778 \uacbd\uc6b0 \ucf24\ub808\uc804\uce58\ud589\ub82c\uc740 \ub2e8\uc21c\ud788 \uc804\uce58\ud589\ub82c \\(A^* = \\,^t\\! A\\)\uc774\ub2e4.<\/p>\n

\ucf24\ub808\uc804\uce58\ud589\ub82c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \uacf1\uc774 \uc815\uc758\ub420 \ub54c, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[(AB)^* = B^* A^* .\\]<\/p>\n

\n

\uc815\ub9ac 6.<\/span>
\n\uc784\uc758\uc758 \\(\\mathbf{w},\\,\\mathbf{z} \\in \\mathbb{C}^n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\langle A\\mathbf{w} \\,\\vert\\, \\mathbf{z} \\rangle = \\langle \\mathbf{w} \\,\\vert\\, A^* \\mathbf{z} \\rangle\\)\uc774\ub2e4. \uc989 \\(A^*\\)\ub294 \\(A\\)\uc758 \uc218\ubc18\ubcc0\ud658\uc774\ub2e4. \ud2b9\ud788 \uc2e4\ud589\ub82c\uc758 \uc218\ubc18\ubcc0\ud658\uc740 \uc804\uce58\ud589\ub82c\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\ub4f1\uc2dd \\(\\langle \\mathbf{w} \\,\\vert\\, \\mathbf{z}\\rangle = \\,^t \\mathbf{w}\\,\\overline{\\mathbf{z}}\\)\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\langle A\\mathbf{w} \\,\\vert\\, \\mathbf{z} \\rangle
\n&= \\,^t (A\\mathbf{w}) \\overline{\\mathbf{z}} = \\,^t \\mathbf{w} \\,^t \\! A \\overline{\\mathbf{z}} \\\\[5pt]
\n&= \\,^t \\mathbf{w} \\overline{A^* \\mathbf{z}} = \\langle \\mathbf{w} \\,\\vert\\, A^* \\mathbf{z} \\rangle . \\tag*{\\(\\blacksquare\\)}
\n\\end{align}\\]\n<\/p>\n<\/div>\n

\uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uac00 \uc870\uac74 \\(T = T^*\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(T\\)\ub97c \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658<\/span>(Hermitian transformation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \ud589\ub82c \\(A\\)\uac00 \uc870\uac74 \\(A = A^*\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(A\\)\ub97c \uc5d0\ub974\ubbf8\ud2b8 \ud589\ub82c<\/span>(Hermitian matrix)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc2e4\ud589\ub82c\uc758 \uacbd\uc6b0 \uc8fc\uc5b4\uc9c4 \uc815\uc0ac\uac01\ud589\ub82c\uc774 \uc5d0\ub974\ubbf8\ud2b8 \ud589\ub82c\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ub300\uce6d\ud589\ub82c\uc778 \uac83\uc774\ub2e4.<\/p>\n

\\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc778 \uacbd\uc6b0 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\langle T(u) \\,\\vert\\,v \\rangle = \\langle u \\,\\vert\\, T(v) \\rangle.\\]
\n\uc815\uaddc\uc9c1\uad50\uae30\uc800\uac00 \uc8fc\uc5b4\uc9c4 \ub0b4\uc801\uacf5\uac04\uc5d0\uc11c\ub294 \uc8fc\uc5b4\uc9c4 \ubcc0\ud658\uc774 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc778\uc9c0 \uc5ec\ubd80\ub97c \ud310\ubcc4\ud560 \ub54c \ud589\ub82c\uc744 \uc774\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.<\/span>
\n\\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(B = \\left\\{ u_1 ,\\, u_2 ,\\, \\cdots ,\\, u_n \\right\\}\\)\uc774 \\(V\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uc774\uba70 \\(T\\)\uac00 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uae30\uc800 \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c \\(A = (a_{ij})\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ud589\ub82c\uc778 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\ub2e4\uc74c \ub450 \ub4f1\uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790.
\n\\[\\begin{gather}
\n\\langle T(u_j) \\,\\vert\\,u_i \\rangle = \\left\\langle\\sum_{k=1}^n a_{kj} u_k \\,\\bigg\\vert\\, u_i \\right\\rangle = \\langle a_{ij} u_i \\,\\vert\\, u_i \\rangle = a_{ij},\\\\[5pt]
\n\\langle u_j \\,\\vert\\, T(u_j)\\rangle = \\left\\langle u_j \\,\\bigg\\vert\\, \\sum_{k=1}^n a_{ki} u_k \\right\\rangle = \\langle u_j \\,\\vert\\, a_{ji} u_j \\rangle = \\overline{a_{ji}}.
\n\\end{gather}\\]
\n\uba85\ubc31\ud788, \ub450 \uac12\uc774 \uac19\uc544\uc9c0\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A = A^*\\)\uc778 \uac83\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\uc815\ub9ac 8.<\/span>
\n\uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc758 \uadfc\uc740 \ud56d\uc0c1 \uc2e4\uc218\uc774\ub2e4. \uc989 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc758 \uace0\uc733\uac12\uc740 \ud56d\uc0c1 \uc2e4\uc218\uc774\ub2e4. \ub354\uc6b1\uc774 \uadf8 \uace0\uc733\uac12\ub4e4\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\ubca1\ud130\ub4e4\uc740 \uc815\uaddc\uc9c1\uad50\uc871\uc744 \uc774\ub8ec\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(T : V \\rightarrow V\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(V\\)\uc5d0 \uc815\uaddc\uc9c1\uad50\uae30\uc800\uac00 \uc8fc\uc5b4\uc838 \uc788\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uadf8 \uae30\uc800\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \uc5d0\ub974\ubbf8\ud2b8 \ud589\ub82c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c \\(A\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ud589\ub82c\uc77c \ub54c \uadf8 \uace0\uc733\uac12\uc774 \uc2e4\uc218\uc784\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4.<\/p>\n

\\(V\\)\uc5d0 \uc8fc\uc5b4\uc9c4 \ub0b4\uc801\uc774 \uc2e4\ub0b4\uc801\uc774\ub4e0 \ubcf5\uc18c\ub0b4\uc801\uc774\ub4e0 \uc0c1\uad00 \uc5c6\uc774 \ud589\ub82c \\(A\\)\ub97c \uc67c\ucabd\uc5d0 \uacf1\ud558\ub294 \uc5f0\uc0b0\uc740 \ubcf5\uc18c\ubca1\ud130\uacf5\uac04 \\(\\mathbb{C}^n\\) \uc704\uc5d0\uc11c\uc758 \ubcc0\ud658\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A\\)\ub97c \\(\\mathbb{C}^n\\) \uc704\uc5d0\uc11c\uc758 \ubcc0\ud658\uc73c\ub85c \uac04\uc8fc\ud558\uace0 \ub17c\uc758\ub97c \uc774\uc5b4\uac00\uc790. \uc774\uc640 \uac19\uc740 \uad00\uc810\uc774 \ud544\uc694\ud55c \uc774\uc720\ub294 \\(A\\)\uc758 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\uac00 \uc874\uc7ac\ud55c\ub2e4\ub294 \uac83\uc744 \ubcf4\uc7a5\ud558\ub824\uba74 \ubcf5\uc18c\uc218 \ubc94\uc704\uc5d0\uc11c \uc0dd\uac01\ud574\uc57c \ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

\\(\\lambda\\)\uac00 \\(A\\)\uc758 \uace0\uc733\uac12\uc774\uace0, \uc774 \uace0\uc733\uac12\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\uac00 \uc601\ubca1\ud130\uac00 \uc544\ub2cc \ubca1\ud130 \\(w\\in\\mathbb{C}^n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\n\\lambda\\langle \\mathbf{w}\\,\\vert\\,\\mathbf{w} \\rangle
\n&= \\langle \\lambda\\mathbf{w} \\,\\vert\\, \\mathbf{w} = \\langle A\\mathbf{w} \\,\\vert\\, \\mathbf{w} \\rangle \\\\[5pt]
\n&= \\langle \\mathbf{w} \\,\\vert\\, A\\mathbf{w} \\rangle = \\langle \\mathbf{w} \\,\\vert\\, \\lambda\\mathbf{w} \\rangle = \\overline{\\lambda} \\langle \\mathbf{w} \\,\\vert\\, \\mathbf{w} \\rangle.
\n\\end{align}\\]
\n\uc5ec\uae30\uc11c \\(\\mathbf{w}\\)\uac00 \uc601\ubca1\ud130\uac00 \uc544\ub2c8\ubbc0\ub85c \\(\\langle \\mathbf{w} \\,\\vert\\,\\mathbf{w}\\rangle \\ne 0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130
\n\\[\\lambda = \\overline{\\lambda}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \uc989 \\(\\lambda\\)\ub294 \uc2e4\uc218\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \\(\\lambda_1\\)\uacfc \\(\\lambda_2\\)\uac00 \\(A\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \uace0\uc733\uac12\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc774\ub4e4 \uace0\uc733\uac12\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub97c \\(\\mathbf{w}_1,\\) \\(\\mathbf{w}_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\n\\lambda_1 \\langle \\mathbf{w}_1 \\,\\vert\\, \\mathbf{w}_2 \\rangle
\n&= \\langle \\lambda_1 \\mathbf{w}_1 \\,\\vert\\, \\mathbf{w}_2\\rangle
\n= \\langle A\\mathbf{w}_1 \\,\\vert\\, \\mathbf{w}_2\\rangle \\\\[5pt]
\n&= \\langle \\mathbf{w}_1 \\,\\vert\\, A\\mathbf{w}_2 \\rangle
\n= \\langle \\mathbf{w}_1 \\,\\vert\\, \\lambda_2 \\mathbf{w}_2 \\rangle
\n= \\lambda_2 \\langle \\mathbf{w}_1 \\,\\vert\\, \\mathbf{w}_2 \\rangle
\n\\end{align}\\]
\n\uc5ec\uae30\uc11c \\(\\lambda_1 \\ne \\lambda_2\\)\uc774\ubbc0\ub85c \\(\\langle \\mathbf{w}_1 \\,\\vert\\, \\mathbf{w}_2 \\rangle = 0\\)\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04\uc5d0\uc11c \uc815\uc758\ub41c \uac00\uc5ed\uc778 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(T\\)\uac00 \\(T^{-1} = T^*\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(T\\)\ub97c \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658<\/span>(unitary transformation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \uc720\ub2c8\ud0c0\ub9ac \uc2e4\ud589\ub82c\uc744 \uc9c1\uad50\ud589\ub82c<\/span>(orthogonal matrix)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc218\ubc18\ubcc0\ud658\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(T\\)\uac00 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\langle T(u) \\,\\vert\\, T(v) \\rangle
\n= \\langle u \\,\\vert\\, T^* T(v) \\rangle
\n= \\langle u\\,\\vert\\, v\\rangle.\\]
\n\uadf8\ub7ec\ubbc0\ub85c \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc740 \uae38\uc774\uc640 \uac01\uc744 \ubcf4\uc874\ud55c\ub2e4. \ub610\ud55c \uc774\ub7ec\ud55c \uba85\uc81c\uc758 \uc5ed\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 9.<\/span>
\n\\(T\\)\uac00 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \uc784\uc758\uc758 \\(u\\in V\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\langle T(u) \\,\\vert\\, T(u) \\rangle = \\langle u\\,\\vert\\, u\\rangle\\tag{11}\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74(\uc989 \\(T\\)\uac00 \uae38\uc774\ub97c \ubcf4\uc874\ud558\uba74) \\(T\\)\ub294 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\ub4f1\uc2dd
\n\\[\\langle T(u+v) \\,\\vert\\, T(u+v) \\rangle = \\langle u \\,\\vert\\, u \\rangle\\]
\n\uc758 \uc591\ubcc0\uc744 \uc804\uac1c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\n\\langle T(u) \\,\\vert\\, T(u) \\rangle + \\langle T(u) \\,\\vert\\, T(v) \\rangle
\n&+ \\langle T(v) \\,\\vert\\, T(u) \\rangle + \\langle T(v) \\,\\vert\\, T(v) \\rangle \\\\[5pt]
\n&= \\langle u\\,\\vert\\,u\\rangle + \\langle u\\,\\vert\\,v\\rangle + \\langle v\\,\\vert\\,u\\rangle + \\langle v\\,\\vert\\,v\\rangle .
\n\\end{align}\\tag{12}\\]
\n(11)\uc744 \uc774\uc6a9\ud558\uc5ec (12)\uc758 \uc591\ubcc0\uc5d0\uc11c \uac19\uc740 \uac12\uc744 \ube7c\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\langle T(u) \\,\\vert\\, T(v) \\rangle + \\langle T(v) \\,\\vert\\, T(u) \\rangle
\n= \\langle u\\,\\vert\\,v\\rangle + \\langle v\\,\\vert\\,u \\rangle.\\tag{13}\\]
\n\uc2e4\ub0b4\uc801\uacf5\uac04\uc758 \uacbd\uc6b0 \uc774 \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130
\n\\[\\langle u\\,\\vert\\, T^* T(v)\\rangle = \\langle T(u) \\,\\vert\\,T(v)\\rangle = \\langle u\\,\\vert\\,v\\rangle\\]
\n\ub97c \uc5bb\ub294\ub2e4. \uc5ec\uae30\uc11c \\(u\\)\uc640 \\(v\\)\ub294 \uc784\uc758\uc758 \ubca1\ud130\uc774\ubbc0\ub85c \\(T^* T = I\\) \uc989 \\(T^* = T^{-1}\\)\uc774\ub2e4.<\/p>\n

\ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc758 \uacbd\uc6b0 (13)\uc758 \uc591\ubcc0\uc740 \ubcf5\uc18c\uc218\uc640 \uadf8 \ucf24\ub808\ubcf5\uc18c\uc218\uc758 \ud569\uc744 \ub098\ud0c0\ub0b8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (13)\uc758 \uc591\ubcc0\uc744 \\(2\\)\ub85c \ub098\ub204\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\operatorname{Re}(\\langle T(u) \\,\\vert\\, T(u) \\rangle) = \\operatorname{Re}(\\langle u\\,\\vert\\,v \\rangle).\\]
\n\uc774 \ub4f1\uc2dd\uc5d0\uc11c \\(u\\)\ub97c \\(\\mathbf{i}u\\)\ub85c \ubc14\uafb8\uace0 \\(\\operatorname{Re}(\\mathbf{i}z) = – \\operatorname{Im}(z)\\)\uc640 \\(T\\)\uc758 \uc120\ud615\uc131\uc744 \uc774\uc6a9\ud558\uc5ec \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\operatorname{Im}(\\langle T(u) \\,\\vert\\, T(u) \\rangle) = \\operatorname{Im}(\\langle u\\,\\vert\\,v \\rangle).\\]
\n\ub530\ub77c\uc11c \\(\\langle u\\,\\vert\\,v\\rangle\\)\uc640 \\(\\langle T(u) \\,\\vert\\,T(v) \\rangle\\)\uc740 \uc2e4\uc218\ubd80\uc640 \ud5c8\uc218\ubd80\uac00 \ubaa8\ub450 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc2e4\ub0b4\uc801 \uacf5\uac04\uc758 \uacbd\uc6b0\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(T^* = T^{-1}\\)\ub97c \uc5bb\ub294\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\ud589\ub82c \\(A \\,\\overline{^t \\! A}\\)\uc758 \\((i,\\,j)\\)-\uc131\ubd84\uc740 \\(A\\)\uc758 \\(i\\)\uc9f8 \ud589\ubca1\ud130\uc640 \\(^t \\! A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\uc758 \ub0b4\uc801\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc774 \uac12\uc740 \\(A\\)\uc758 \\(i\\)\uc9f8 \uc5f4\ubca1\ud130\uc640 \\(A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\uc758 \ub0b4\uc801\uacfc \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A \\,\\overline{^t \\! A}\\)\uac00 \\(I_n = (\\delta_{ij})\\)\uc640 \uac19\uc744 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A\\)\uc758 \ud589\ubca1\ud130\ub4e4\uc774 \uc815\uaddc\uc9c1\uad50\uc871\uc744 \uc774\ub8e8\ub294 \uac83\uc774\ub2e4.<\/p>\n

\uc720\ub2c8\ud0c0\ub9ac \ud589\ub82c\uc758 \uc804\uce58\ud589\ub82c \ub610\ud55c \uc720\ub2c8\ud0c0\ub9ac \ud589\ub82c\uc774\ubbc0\ub85c, \uc774\uc640 \uac19\uc740 \uacb0\uacfc\ub294 \\(A\\)\uc758 \uc5f4\ubca1\ud130\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 10.<\/span>
\n\uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc758 \uace0\uc733\uac12\uc758 \ud06c\uae30(\uc808\ub313\uac12)\ub294 \ud56d\uc0c1 \\(1\\)\uc774\ub2e4. \ub354\uc6b1\uc774 \uc11c\ub85c \ub2e4\ub978 \uace0\uc733\uac12\ub4e4\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub4e4\uc740 \uc9c1\uad50\ubca1\ud130\uc871\uc744 \uc774\ub8ec\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(T: V \\rightarrow V\\)\uac00 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\uace0 \\(\\lambda\\)\uac00 \uc774 \ubcc0\ud658\uc758 \uace0\uc733\uac12\uc774\uba70 \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130 \\(u\\)\uac00 \uc601\ubca1\ud130\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\langle u \\,\\vert\\, u \\rangle = \\langle T(u) \\,\\vert\\, T(u) \\rangle = \\langle \\lambda u \\,\\vert\\, \\lambda u \\rangle = \\lambda \\overline{\\lambda} \\langle u\\,\\vert\\, u \\rangle\\]
\n\uc774\ubbc0\ub85c \\(\\lambda\\overline{\\lambda} = 1\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\lambda\\)\uc758 \uc808\ub313\uac12\uc740 \\(1\\)\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \\(\\lambda_1\\)\uacfc \\(\\lambda_2\\)\uac00 \\(T\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \uace0\uc733\uac12\uc774\uace0 \\(u_1\\)\uacfc \\(u_2\\)\uac00 \uc774\ub4e4\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\n\\langle u_1\\,\\vert\\, u_2 \\rangle
\n&= \\langle T(u_1 ) \\,\\vert\\, T(u_2) \\rangle
\n= \\langle \\lambda_1 u_1 \\,\\vert\\, \\lambda_2 u_2 \\rangle \\\\[5pt]
\n&= \\lambda_1 \\overline{\\lambda_2} \\langle u_1 \\,\\vert\\, u_2 \\rangle
\n= \\frac{\\lambda_1}{\\lambda_2} \\langle u_1 \\,\\vert\\, u_2 \\rangle .
\n\\end{align}\\]
\n\uadf8\ub7f0\ub370 \\(\\lambda_1 \\ne \\lambda_2\\)\uc774\ubbc0\ub85c \\(\\langle u_1\\,\\vert\\, u_2 \\rangle = 0\\)\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

<\/p>\n

\uc2a4\ud399\ud2b8\ub7fc \ubd84\ud574<\/h2>\n
\n

\ubcf4\uc870\uc815\ub9ac 11.<\/span>
\n\\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\uac70\ub098 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(u\\)\uac00 \\(T\\)\uc758 \uace0\uc720\ubca1\ud130\uc774\uace0 \\(v\\)\uac00 \\(u\\)\uc640 \uc218\uc9c1\uc778 \ubca1\ud130\uc774\uba74, \\(T(v)\\)\ub3c4 \\(u\\)\uc640 \uc218\uc9c1\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\uace0 \\(\\lambda\\)\uac00 \\(T\\)\uc758 \uace0\uc733\uac12\uc774\uba70 \\(u\\)\uac00 \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\lambda\\langle u\\,\\vert\\,v\\rangle = \\langle \\lambda u \\,\\vert\\,v \\rangle = \\langle T(u) \\,\\vert\\, v \\rangle = \\langle u \\,\\vert\\, T(v) \\rangle.\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(\\langle u \\,\\vert\\,v \\rangle = 0\\)\uc77c \ub54c \\(\\langle u \\,\\vert\\, T(v) \\rangle = 0\\)\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \\(T\\)\uac00 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\uace0 \\(\\lambda\\)\uac00 \\(T\\)\uc758 \uace0\uc733\uac12\uc774\uba70 \\(u\\)\uac00 \\(\\lambda\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub77c\uace0 \ud558\uc790. \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc758 \uace0\uc733\uac12\uc758 \ud06c\uae30\ub294 \ud56d\uc0c1 \\(1\\)\uc774\ubbc0\ub85c \\(\\lambda\\)\ub294 \\(0\\)\uc774 \uc544\ub2c8\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\langle u \\,\\vert\\, v \\rangle = \\langle T(u) \\,\\vert\\, T(v) \\rangle = \\langle \\lambda u \\,\\vert\\, T(v) \\rangle = \\lambda \\langle u \\,\\vert\\, T(v) \\rangle.\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(\\langle u \\,\\vert\\,v \\rangle = 0\\)\uc77c \ub54c \\(\\langle u \\,\\vert\\, T(v) \\rangle = 0\\)\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\\(T : V \\rightarrow V\\)\uac00 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\uace0 \\(W\\)\uac00 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(w\\in W\\)\uc5d0 \ub300\ud558\uc5ec \\(T(w) \\in W\\)\uc774\uba74 \u2018\\(W\\)\ub294 \\(T\\)\uc5d0 \ub300\ud574 \ubd88\ubcc0\uc774\ub2e4\u2019 \ub610\ub294 \u2018\\(W\\)\ub294 \\(T\\)-\ubd88\ubcc0<\/span>\uc774\ub2e4(T-invariant)\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774 \uacbd\uc6b0 \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(W\\)\ub85c \ucd95\uc18c\ud55c \uc81c\ud55c\ud568\uc218 \\(T \\vert_W\\)\ub294 \\(W\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<\/p>\n

\\(W\\)\uac00 \\(T\\)-\ubd88\ubcc0\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \n
  1. \\(T\\vert_W\\)\uc758 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\ub294 \\(T\\)\uc758 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\uc640 \ub3d9\uc77c\ud558\ub2e4.<\/li>\n
  2. \\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\uba74 \\(T\\vert_W\\)\ub3c4 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(V\\)\uc5d0\uc11c \\[\\langle T(u) \\,\\vert\\, v \\rangle = \\langle u \\,\\vert\\, T(v)\\rangle\\]\uac00 \uc131\ub9bd\ud558\uba74, \uc774 \ub4f1\uc2dd\uc740 \\(W\\)\uc5d0\uc11c\ub3c4 \uc131\ub9bd\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n
  3. \\(T\\)\uac00 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\uba74 \\(T\\vert_W\\)\ub3c4 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(V\\)\uc5d0\uc11c \\[\\langle T(u) \\,\\vert\\, T(v) \\rangle = \\langle u \\,\\vert\\, v\\rangle\\]\uac00 \uc131\ub9bd\ud558\uba74, \uc774 \ub4f1\uc2dd\uc740 \\(W\\)\uc5d0\uc11c\ub3c4 \uc131\ub9bd\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<\/ol>\n
    \n

    \uc815\ub9ac 12. (\uc2a4\ud399\ud2b8\ub7fc \ubd84\ud574)<\/span><\/p>\n

    \\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(T\\)\uac00 \\(V\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(T\\)\uac00 \ub2e4\uc74c \ub450 \uc870\uac74 \uc911 \ud558\ub098\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790.<\/p>\n

      \n
    1. \\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\uace0 \\(V\\)\uac00 \\(\\mathbb{R}\\) \ub610\ub294 \\(\\mathbb{C}\\) \uc704\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<\/li>\n
    2. \\(T\\)\uac00 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\uace0 \\(V\\)\uac00 \\(\\mathbb{C}\\) \uc704\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<\/li>\n<\/ol>\n

      \uadf8\ub7ec\uba74 \\(T\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 \uc815\uaddc\uc9c1\uad50\uace0\uc720\uae30\uc800\uac00 \uc874\uc7ac\ud55c\ub2e4. \ud2b9\ud788 \\(T\\)\ub294 \ub300\uac01\ud654 \uac00\ub2a5\ud558\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85.<\/p>\n

      \\(T\\)\uc758 \ubaa8\ub4e0 \uace0\uc733\uac12\uc740 \\(V\\)\uac00 \uc815\uc758\ub41c \uccb4\uc5d0 \uc18d\ud55c\ub2e4. (i)\uc758 \uacbd\uc6b0 \\(T\\)\uac00 \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\ubbc0\ub85c \\(T\\)\uc758 \ubaa8\ub4e0 \uace0\uc733\uac12\uc740 \uc2e4\uc218\uc774\ub2e4. (ii)\uc758 \uacbd\uc6b0 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc758 \ubaa8\ub4e0 \uadfc\uc774 \uccb4 \\(\\mathbb{C}\\)\uc5d0 \uc18d\ud55c\ub2e4. \\(\\dim(V)= n\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

      \\(\\lambda_1\\)\uc774 \\(T\\)\uc758 \uace0\uc733\uac12\uc774\uace0 \\(u_1\\)\uc774 \\(\\lambda_1\\)\uc5d0 \ub530\ub974\ub294 \uace0\uc720\ubca1\ud130\ub77c\uace0 \ud558\uc790. \\(u_1\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \ubd80\ubd84\uacf5\uac04 \\(W\\)\ub294 \\(1\\)\ucc28\uc6d0\uc774\uba70, \\(W\\)\uc758 \uc9c1\uad50\uc5ec\uacf5\uac04 \\(W^\\bot\\)\uc740 \\(n-1\\)\ucc28\uc6d0\uc774\ub2e4. \uba85\ubc31\ud788 \\(T\\)\uac00 \\(W\\)\uc758 \ubaa8\ub4e0 \ubca1\ud130\ub97c \\(W\\)\uc758 \ubca1\ud130\uc5d0 \ub300\uc751\uc2dc\ud0a4\ubbc0\ub85c, \\(T\\)\ub294 \\(W^\\bot\\)\uc758 \ubaa8\ub4e0 \ubca1\ud130\ub97c \\(W^\\bot\\)\uc758 \ubca1\ud130\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(W^\\bot\\)\ub294 \\(T\\)\uc5d0 \ub300\ud574 \ubd88\ubcc0\uc778 \ubd80\ubd84\uacf5\uac04\uc774\ub2e4.<\/p>\n

      \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(W^\\bot\\)\uc73c\ub85c \uc81c\ud55c\ud558\uc5ec\ub3c4 \\(T\\)\ub294 (i)\uc5d0 \uc758\ud558\uc5ec \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uc774\uac70\ub098, (ii)\uc5d0 \uc758\ud558\uc5ec \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc774\ub2e4. \uadf8\ub9ac\uace0 \\(T\\)\uc758 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\ub294 \\(T\\vert_{W^\\bot}\\)\uc758 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\uc774\ub2e4. \\(T\\)\uc758 \uace0\uc733\uac12 \\(\\lambda_2\\)\uc640 \uace0\uc720\ubca1\ud130 \\(u_2\\)\ub97c \ud0dd\ud558\uace0, \ub2e4\uc2dc \uc55e\uc758 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \\(u_2\\)\uc640 \uc218\uc9c1\uc778 \ubca1\ud130\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \\(W\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n

      \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \\(T\\)\uc758 \uace0\uc733\uac12 \\(\\lambda_3 ,\\) \\(\\cdots,\\) \\(\\lambda_n\\)\uacfc \uace0\uc720\ubca1\ud130 \\(u_3,\\) \\(\\cdots,\\) \\(u_n\\)\uc744 \uc5bb\ub294\ub2e4. \uc774 \uace0\uc720\ubca1\ud130\ub4e4\uc744 \uac01\uac01\uc758 \ud06c\uae30\ub85c \ub098\ub204\uc5b4 \uc8fc\uba74 \\(T\\)\uc758 \uace0\uc720\ubca1\ud130\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc815\uaddc\uc9c1\uad50 \uace0\uc720\uae30\uc800\ub97c \uc5bb\ub294\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n","protected":false},"excerpt":{"rendered":"

      \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\uc758 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf4\uace0 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \uace0\uc733\uac12\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubd05\ub2c8\ub2e4. \ub610\ud55c \uc5d0\ub974\ubbf8\ud2b8 \ubcc0\ud658\uacfc \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658\uc758 \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \uc2a4\ud399\ud2b8\ub7fc \ubd84\ud574 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubd05\ub2c8\ub2e4. \\( \\newcommand{\\parallelsym}{\\mathbin{\\!\/\\mkern-5mu\/\\!}} \\) \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\uc758 \ub73b \\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T : V \\rightarrow V\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc2a4\uce7c\ub77c \\(\\lambda \\in K\\)\uc640 \uc601\ubca1\ud130\uac00 \uc544\ub2cc \ubca1\ud130 \\(v\\in V\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(T(v) = \\lambda v\\) \uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(\\lambda\\)\ub97c \\(T\\)\uc758 \uace0\uc733\uac12(eigenvalue)\uc774\ub77c\uace0…<\/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":[479,467,468,466,465,470,469,474,480,472,473,478,477,476,475,471],"class_list":["post-5453","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-t-","tag-467","tag-468","tag-466","tag-465","tag-470","tag-469","tag-474","tag-480","tag-472","tag-473","tag-478","tag-477","tag-476","tag-475","tag-471"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5453","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=5453"}],"version-history":[{"count":68,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5453\/revisions"}],"predecessor-version":[{"id":6114,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5453\/revisions\/6114"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}