{"id":5721,"date":"2020-11-06T10:18:21","date_gmt":"2020-11-06T01:18:21","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5721"},"modified":"2020-11-09T15:45:28","modified_gmt":"2020-11-09T06:45:28","slug":"linear-algebra-cayley-hamilton-theorem-and-triangulation-of-endomorphisms","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-cayley-hamilton-theorem-and-triangulation-of-endomorphisms\/","title":{"rendered":"Cayley-Hamilton \uc815\ub9ac\uc640 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \uc0bc\uac01\ud589\ub82c \ud45c\ud604"},"content":{"rendered":"

\uc815\uc0ac\uac01\ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \uc774\uc6a9\ud55c \ud765\ubbf8\ub85c\uc6b4 \ub4f1\uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\(A\\)\uac00 \uc774\ucc28\uc7a5\uc0ac\uac01\ud589\ub82c\uc774\uace0
\n\\[A = \\left(\\begin{array}{cc} a & b \\\\ c & d \\end{array}\\right)\\]
\n\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[A^2 – (a+d)A + (ad-bc)I_2 = O.\\]
\n\\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \\(p(t)\\)\ub77c\uace0 \ud558\uace0 \\(t=A\\)\ub97c \ub300\uc785\ud568\uc73c\ub85c\uc368 \uc704 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uac04\ub2e8\ud558\uac8c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
\n\\[p(A) = O.\\]
\n\uc774 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc740 \uc6b0\uc5f0\uc774 \uc544\ub2c8\uba70, Cayley-Hamilton \uc815\ub9ac\uc758 \uacb0\uacfc\uc774\ub2e4.<\/p>\n

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc758 \uc131\uc9c8\uacfc \\(T\\)-\ubd88\ubcc0 \uacf5\uac04\uc758 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uc774\uc5b4\uc11c Cayley-Hamilton \uc815\ub9ac\uc640 \uadf8 \uc99d\uba85\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub610\ud55c Cayley-Hamilton \uc815\ub9ac\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud560 \uc218 \uc788\ub294 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \uc0bc\uac01\ud589\ub82c \ucd95\uc57d \uc815\ub9ac\ub3c4 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\n\\[
\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}
\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}
\n\\newcommand{\\proj}{{\\operatorname{proj}}}
\n\\newcommand{\\adj}{{\\operatorname{adj}}}
\n\\]\n<\/div>\n

<\/p>\n

Cayley-Hamilton \uc815\ub9ac<\/h2>\n

\\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\ub4e4\uc758 \ubaa8\uc784 \\(\\Hom (V,\\,V)\\)\ub294 \\(k\\)-\ub300\uc218\uac00 \ub41c\ub2e4. \uc989 \\(\\Hom (V,\\,V)\\)\uc5d0\ub294 \uc2a4\uce7c\ub77c\uacf1, \ud568\uc218\ud569, \ud568\uc218\ud569\uc131\uc774\ub77c\ub294 \uc138 \uac1c\uc758 \uc5f0\uc0b0\uc774 \uc8fc\uc5b4\uc838 \uc788\uc73c\uba70, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \n
  1. \uc2a4\uce7c\ub77c\uacf1\uacfc \ud568\uc218\ud569\uc5d0 \ub300\ud558\uc5ec \\(\\Hom (V,\\,V)\\)\ub294 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub2e4.<\/li>\n
  2. \ud568\uc218\ud569\uacfc \ud568\uc218\ud569\uc131\uc5d0 \ub300\ud558\uc5ec \\(\\Hom (V,\\,V)\\)\ub294 \ub2e8\uc704\uc6d0\uc744 \uac16\ub294 \ud658\uc774\ub2e4.<\/li>\n
  3. \uc2a4\uce7c\ub77c \uacf1\uacfc \ud568\uc218 \ud569\uc131\uc740 \uc11c\ub85c \uad50\ud658 \uac00\ub2a5\ud558\ub2e4. \uc989 \\(\\lambda\\in K\\)\uc640 \\(T_1,\\,T_2 \\in \\Hom(V,\\,V)\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[(\\lambda T_1 )T_2 = T_1 (\\lambda T_2 ) = \\lambda (T_1 T_2)\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(T_1 T_2\\)\ub294 \ud569\uc131 \\(T_1 \\circ T_2\\)\ub97c \uc758\ubbf8\ud55c\ub2e4.<\/li>\n<\/ol>\n

    \uc0c1\uc218\uc640 \uacc4\uc218\uac00 \\(K\\)\uc5d0 \uc18d\ud558\uace0 \\(t\\)\ub97c \ubcc0\uc218\ub85c \uac16\ub294 \ubaa8\ub4e0 \ub2e4\ud56d\uc2dd\ub4e4\uc758 \ubaa8\uc784\uc744 \\(K[t]\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(K[t]\\)\ub294 \\(K\\)-\ub300\uc218\uc774\ub2e4.<\/p>\n

    \\(p(t)\\in K[t]\\)\uc774\uace0
    \n\\[p(t) = \\sum_{j=1}^r \\alpha_j t^j \\quad (\\alpha_0 ,\\, \\alpha_1 ,\\, \\cdots ,\\, \\alpha_r \\in K )\\]
    \n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubcc0\ud658 \\(T\\in\\Hom(V,\\,V)\\)\uc5d0 \ub300\ud558\uc5ec \\(p(T)\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud589\ub82c\uc774\ub2e4.
    \n\\[p(T) = \\sum_{j=0}^r \\alpha_j T^j = \\alpha_r T^r + \\alpha_{r-1}T^{r-1} + \\cdots + \\alpha_1 T + \\alpha_0 1_V .\\]
    \n\uc5ec\uae30\uc11c \\(T^j\\)\ub294 \\(j\\)\uac1c\uc758 \\(T\\)\ub97c \ud569\uc131\ud55c \uac83\uc774\uba70, \\(T^0\\)\uc740 \ud56d\ub4f1\ubcc0\ud658\uc744 \ub098\ud0c0\ub0b8\ub2e4. \\(T\\)\ub97c \uace0\uc815\uc2dc\ucf30\uc744 \ub54c \ub300\uc751
    \n\\[K[t] \\,\\mapsto\\, \\Hom(V,\\,V) ,\\quad p\\,\\mapsto\\,p(T)\\]
    \n\ub294 \uba85\ubc31\ud788 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uc774\ub7ec\ud55c \ub300\uc751\uc740 \uc120\ud615\uc774\ub77c\ub294 \ud2b9\uc9d5 \uc678\uc5d0\ub3c4 \ub354 \ub9ce\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4. \\(T\\)\uc758 \uac70\ub4ed\uc81c\uacf1\uc758 \uacf1\uc5d0\uc11c \uc9c0\uc218\ubc95\uce59\uc774 \uc131\ub9bd\ud558\uae30 \ub54c\ubb38\uc5d0, \\(K[t]\\)\uc758 \uc6d0\uc18c \\(p,\\) \\(q\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
    \n\\[p(T) q(T) = pq(T)\\]
    \n\uc5ec\uae30\uc11c \uc6b0\ubcc0\uc740 \ub450 \ub2e4\ud56d\uc2dd \\(p\\)\uc640 \\(q\\)\ub97c \uacf1\ud55c \ub4a4 \ubcc0\uc218\uc5d0 \\(T\\)\ub97c \ub300\uc785\ud558\uc5ec \uacc4\uc0b0\ud55c \uac83\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(T\\)\uc5d0\uc11c \ub2e4\ud56d\uc2dd\uc744 \uacc4\uc0b0\ud558\ub294 \uc5f0\uc0b0\uc740 \\(K\\)-\ub300\uc218\uc758 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<\/p>\n

    \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc758 \uc120\ud615\ubcc0\ud658\uc740 \ud589\ub82c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc73c\ubbc0\ub85c, \uc9c0\uae08\uae4c\uc9c0 \ub17c\uc758\ud55c \ub0b4\uc6a9\uc740 \ud589\ub82c\uacf5\uac04 \\(M_n (K)\\)\uc758 \uc6d0\uc18c \\(A\\)\uc5d0\ub3c4 \uadf8\ub300\ub85c \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc989 \\(p(t)\\in K[t]\\)\uc774\uace0 \\(A\\in M_n(K)\\)\uc77c \ub54c
    \n\\[p(A) = \\sum_{j=0}^r \\alpha_j A^j = \\alpha_r A^r + \\alpha_{r-1} A^{r-1} + \\cdots + \\alpha_1 A + \\alpha_0 I_n\\]
    \n\uc774\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 1. (Cayley-Hamilton \uc815\ub9ac)<\/span><\/p>\n

    \\(V\\)\uac00 \\(K\\) \uc704\uc5d0\uc11c\uc758 \\(n\\)\ucc28\uc6d0 \uacf5\uac04\uc774\uace0 \\(n\\ge 1\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(T\\in \\Hom(V,\\,V)\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(p(t)\\)\uc774\uba74 \\(p(T)=0\\)\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c, \\(A\\in M_n(K)\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(p(t)\\)\uc774\uba74 \\(p(A) = O\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \uc815\ub9ac 1\uc744 \uc99d\uba85\ud558\uae30 \uc704\ud574\uc11c\ub294 \uba87 \uac1c\uc758 \ubcf4\uc870\uc815\ub9ac\uac00 \ud544\uc694\ud558\ub2e4. \uadf8 \ubcf4\uc870\uc815\ub9ac\ub97c \ub3c4\uc785\ud558\uae30 \uc804\uc5d0 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

    \n

    \ubcf4\uae30 1.<\/span>
    \n\\(\\mathbb{R}^2\\)\uc5d0\uc11c \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \uc2dc\uacc4\ubc14\ub298 \ubc18\ub300\ubc29\ud5a5\uc73c\ub85c \\(\\pi\/2\\)\ub9cc\ud07c \ud68c\uc804\uc2dc\ud0a4\ub294 \ubcc0\ud658\uc758 \ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[A = \\left(\\begin{array}{cr} 0 & -1 \\\\ 1 & 0 \\end{array}\\right).\\]
    \n\uc774 \ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \\(p(t) = t^2 + 1\\)\uc774\ub2e4. Cayley-Hamilton \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(A^2 + I_2\\)\ub294 \uc601\ud589\ub82c\uc774\ub2e4. \uc2e4\uc81c\ub85c \uacc4\uc0b0\ud574 \ubcf4\uba74
    \n\\[A^2 = \\left(\\begin{array}{rr} -1 & 0 \\\\ 0 & -1 \\end{array}\\right)\\]
    \n\uc774\ubbc0\ub85c \\(A^2 + I_2 = O\\)\uac00 \uc131\ub9bd\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc774 \ud655\uc778\ub41c\ub2e4.\n<\/p>\n<\/div>\n

    \uba87 \uac00\uc9c0 \ubcf4\uc870\uc815\ub9ac<\/h2>\n

    \\(T : V \\rightarrow V\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\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, \\(W\\)\ub294 \\(T\\)\uc5d0 \ub300\ud574 \ubd88\ubcc0<\/span>\uc774\ub2e4, \ub610\ub294 \\(W\\)\ub294 \u2018\\(T\\)-\ubd88\ubcc0<\/span>(T-invariant)\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

    \uc989 \\(W\\)\uac00 \\(T\\)-\ubd88\ubcc0\uc774\ub77c\ub294 \uac83\uc740 \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(W\\)\ub85c \ucd95\uc18c\ud558\uc600\uc744 \ub54c \\(T\\vert_W\\)\uac00 \\(W\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774 \ub418\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \\(V\\) \uc790\uc2e0\uacfc \uc790\uba85\ud55c \uacf5\uac04 \\(\\left\\{ \\mathbf{0} \\right\\}\\)\ub294 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\ubbc0\ub85c, \uc774 \ub450 \uacf5\uac04\uc740 \uc784\uc758\uc758 \uc120\ud615\ubcc0\ud658 \\(T:V \\rightarrow V\\)\uc5d0 \ub300\ud558\uc5ec \\(T\\)-\ubd88\ubcc0\uc774\ub2e4. \\(T\\)\uc758 \uace0\uc733\uac12 \\(\\lambda\\)\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\uacf5\uac04 \\(W\\) \uc5ed\uc2dc \\(T\\)-\ubd88\ubcc0\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc784\uc758\uc758 \\(w\\in W\\)\uc5d0 \ub300\ud558\uc5ec \\(T(w) = \\lambda w \\in W\\)\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

    \n

    \ubcf4\uc870\uc815\ub9ac 2.<\/span>
    \n\\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T : V \\rightarrow V\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\uba70 \\(n > 1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04 \uc911\uc5d0\uc11c \uc790\uba85\ud55c \uacf5\uac04\uacfc \\(V\\) \uc790\uc2e0\uc744 \uc81c\uc678\ud55c \uc5b4\ub290\uac83\ub3c4 \\(T\\)-\ubd88\ubcc0\uc774 \ub418\uc9c0 \ubabb\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc758 \uae30\uc800 \\(B\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \uc801\ub2f9\ud55c \uc2a4\uce7c\ub77c \\(\\alpha_0 ,\\) \\(\\cdots,\\) \\(\\alpha_{n-1}\\)\uc5d0 \ub300\ud558\uc5ec \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub09c\ub2e4.
    \n\\[A = \\left(\\begin{array}{ccccccc}
    \n0 & 0 & 0 & \\cdots & 0 & 0 & \\alpha_0 \\\\
    \n1 & 0 & 0 & \\cdots & 0 & 0 & \\alpha_1 \\\\
    \n0 & 1 & 0 & \\cdots & 0 & 0 & \\alpha_2 \\\\
    \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\
    \n0 & 0 & 0 & \\cdots & 1 & 0 & \\alpha_{n-2} \\\\
    \n0 & 0 & 0 & \\cdots & 0 & 1 & \\alpha_{n-1}
    \n\\end{array}\\right).\\]
    \n\ub354\uc6b1\uc774 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[p(t) = t^n – \\alpha_{n-1} t^{n-1} – \\alpha_{n-2} t^{n-2} – \\cdots – \\alpha_1 t – \\alpha_0 .\\]\n<\/p>\n<\/div>\n

    \n

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

    \\(v_0 \\in V\\)\uac00 \uc601\ubca1\ud130\uac00 \uc544\ub2cc \ubca1\ud130\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(j=0 ,\\, 1 ,\\, \\cdots ,\\, n-2\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[v_{j+1} = T(v_j)\\]
    \n\ub77c\uace0 \uc815\uc758\ud558\uc790. \uc774\uc81c \\(v_0,\\) \\(v_1 ,\\) \\(\\cdots ,\\) \\(v_{n-1}\\)\uc774 \uc77c\ucc28\ub3c5\ub9bd\uc774\uba70, \\(V\\)\uc758 \uae30\uc800 \\(B\\)\ub97c \ud615\uc131\ud568\uc744 \ubcf4\uc77c \uac83\uc774\ub2e4. \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(v_0,\\) \\(v_1 ,\\) \\(\\cdots ,\\) \\(v_{n-1}\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(v_0 ,\\) \\(\\cdots ,\\) \\(v_j\\)\uac00 \uc77c\ucc28\uc885\uc18d\uc778 \uac00\uc7a5 \uc791\uc740 \\(j\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4. \\(0 < j \\le n-1\\)\uc774\uace0 \\(v_j\\)\uac00 \uc77c\ucc28\uc885\uc18d \uad00\uacc4\uc2dd\uc5d0\uc11c \ubc18\ub4dc\uc2dc \uc0ac\uc6a9\ub418\uc5b4\uc57c \ud558\ubbc0\ub85c \\(v_j\\)\ub294 \\(W = \\operatorname{Span}(v_0 ,\\, \\cdots ,\\, v_{j-1})\\)\uc5d0 \uc18d\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(T(v_0 ),\\) \\(\\cdots,\\) \\(T(v_{j-1})\\)\uc774 \ubaa8\ub450 \\(W\\)\uc5d0 \uc18d\ud558\uba70, \\(W\\)\ub294 \\(T\\)-\ubd88\ubcc0\uc778 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774 \ub41c\ub2e4. \ub354\uc6b1\uc774 \\(v_0 \\in W\\)\uc774\ubbc0\ub85c \\(W\\)\ub294 \uc790\uba85\ud55c \uacf5\uac04\uc774 \uc544\ub2c8\uba70 \\(W\\)\ub294 \\(n\\)\uac1c \ubbf8\ub9cc\uc758 \ubca1\ud130\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ubbc0\ub85c \\(V\\)\uc758 \uc9c4\ubd80\ubd84\uacf5\uac04\uc774\ub2e4. \uc774\uac83\uc740 \uadf8\ub7ec\ud55c \ubd80\ubd84\uacf5\uac04\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\ub294 \uc815\ub9ac\uc758 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n

    \uc774\uc81c \\(v_0,\\) \\(v_1 ,\\) \\(\\cdots ,\\) \\(v_{n-1}\\)\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569 \\(B\\)\ub294 \\(V\\)\uc758 \uae30\uc800\uc774\ub2e4. \uc774\uc81c \uc801\ub2f9\ud55c \uc2a4\uce7c\ub77c\ub4e4 \\(\\alpha_j \\in K\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[T(v_j) = v_{j+1} \\,(0\\le j \\le n-1) \\,\\text{ and }\\, T(v_{n-1}) = \\sum_{j=0}^{n-1} \\alpha_j v_j\\]
    \n\uc774\ubbc0\ub85c, \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c \\(A\\)\ub294 \uc815\ub9ac\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \uac83\uacfc \uc77c\uce58\ud55c\ub2e4.<\/p>\n

    \uc774\uc81c \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc758 \ud615\ud0dc\ub97c \ubc1d\ud788\uae30 \uc704\ud558\uc5ec \ub2e4\uc74c\uc744 \uacc4\uc0b0\ud574\uc57c \ud55c\ub2e4.
    \n\\[p(t) = \\det\\left(\\begin{array}{rrccrrc}
    \nt & 0 & 0 & \\cdots & 0 & 0 & -\\alpha_0 \\\\
    \n-1 & t & 0 & \\cdots & 0 & 0 & -\\alpha_1 \\\\
    \n0 & -1 & t & \\cdots & 0 & 0 & -\\alpha_2 \\\\
    \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\
    \n0 & 0 & 0 & \\cdots & -1 & t & -\\alpha_{n-2} \\\\
    \n0 & 0 & 0 & \\cdots & 0 & -1 & t-\\alpha_{n-1}
    \n\\end{array}\\right).\\]
    \n\uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc790. \\(n=2\\)\uc77c \ub54c\ub294
    \n\\[\\det\\left(\\begin{array}{rc} t & -\\alpha_0 \\\\ -1 & t-\\alpha_1 \\end{array}\\right) = t^2 – \\alpha t – \\alpha_0 \\]
    \n\uc774\ubbc0\ub85c \uc815\ub9ac\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \\(n > 2\\)\uc77c \ub54c\ub294 \uccab\uc9f8 \uc5f4\uc744 \uae30\uc900\uc73c\ub85c \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud558\uc790. \uccab\uc9f8 \uc5f4\uc5d0\uc11c \\(0\\)\uc774 \uc544\ub2cc \uc131\ubd84\uc740 \ub450 \uac1c \ubfd0\uc774\ubbc0\ub85c, \uc804\uac1c\ud55c \uacb0\uacfc \ub450 \uac1c\uc758 \ud56d\uc774 \ub098\ud0c0\ub09c\ub2e4. \uadf8 \uc911 \uccab\uc9f8 \ud56d<\/span>\uc740
    \n\\[t\\cdot \\det\\left(\\begin{array}{rrccrrc}
    \nt & 0 & 0 & \\cdots & 0 & 0 & -\\alpha_1 \\\\
    \n-1 & t & 0 & \\cdots & 0 & 0 & -\\alpha_2 \\\\
    \n0 & -1 & t & \\cdots & 0 & 0 & -\\alpha_3 \\\\
    \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\
    \n0 & 0 & 0 & \\cdots & -1 & t & -\\alpha_{n-2} \\\\
    \n0 & 0 & 0 & \\cdots & 0 & -1 & t-\\alpha_{n-1}
    \n\\end{array}\\right)\\]
    \n\uc774\uba70, \uc774 \uc2dd\uc744 \uc804\uac1c\ud55c \uacb0\uacfc\ub294 \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[t(t^{n-1} – \\alpha_{n-1} t^{n-2} – \\cdots – \\alpha_1 ) = t^n – \\alpha_{n-1} t^{n-1} – \\cdots – \\alpha_1 t.\\tag{*}\\]
    \n\uc774\uc81c \ub458\uc9f8 \ud56d<\/span>\uc744 \uacc4\uc0b0\ud574\uc57c \ud55c\ub2e4. \ub458\uc9f8 \ud56d\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc2dd\uc73c\ub85c \ub098\ud0c0\ub09c\ub2e4.
    \n\\[ – \\alpha_0 (-1)^{n+1} \\det
    \n\\left(\\begin{array}{rcrccr}
    \n-1 & t & 0 & \\cdots & 0 & 0 \\\\
    \n0 & -1 & t & \\cdots & 0 & 0 \\\\
    \n0 & 0 & -1 & \\cdots & 0 & 0 \\\\
    \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\
    \n0 & 0 & 0 & \\cdots & -1 & t \\\\
    \n0 & 0 & 0 & \\cdots & 0 & -1
    \n\\end{array}\\right) = -\\alpha_0.\\tag{**}\\]
    \n\uc774\ub85c\uc368 (*)\uacfc (**)\ub97c \ub354\ud558\uba74 \uc815\ub9ac\uc758 \uc2dd\uc744 \uc5bb\ub294\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \ubcf4\uc870\uc815\ub9ac 2\uc758 \ub450 \ubc88\uc9f8 \ub4f1\uc2dd\uc740 \\(1\\)\ucc28\uc6d0\uc5d0\uc11c\ub3c4 \uc131\ub9bd\ud55c\ub2e4. \uc989 \\(T\\)\uac00 \uc2a4\uce7c\ub77c \\(\\alpha_0\\)\ub97c \uacf1\ud558\ub294 \uc120\ud615\ubcc0\ud658\uc77c \ub54c, \uc774 \ubcc0\ud658\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc740 \\(T – \\alpha_0\\)\uc774\ub2e4.<\/p>\n

    \n

    \ubcf4\uc870\uc815\ub9ac 3.<\/span>
    \n\\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(n\\ge 2\\)\uc774\uba70 \\(T:V \\rightarrow V\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04 \uc911 \\(T\\)-\ubd88\ubcc0\uc778 \uac83\uc740 \\(V\\) \uc790\uc2e0\uacfc \uc790\uba85\ud55c \ubd80\ubd84\uacf5\uac04 \uc678\uc5d0\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(p(t)\\)\uac00 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774\uba74 \\(p(T) =O\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

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

    \\(v_j\\)\uc640 \\(\\alpha_j\\)\uac00 \ubcf4\uc870\uc815\ub9ac 2\uc758 \uc99d\uba85\uc5d0\uc11c\uc640 \uac19\uc740 \uac83\uc774\ub77c\uace0 \ud558\uc790. (\ub2e8, \\(j =0,\\) \\(1,\\) \\(\\cdots,\\) \\(n-1.\\)) \uadf8\ub7ec\uba74 \\(0\\le j\\le n-1\\)\uc77c \ub54c
    \n\\[v_j = T^j (v_0 ) \\,\\text{ and }\\, T(v_{n-1}) = \\sum_{j=0}^{n-1}\\alpha_j v_j\\]
    \n\uc774\ub2e4. \ubcf4\uc870\uc815\ub9ac 2\ub97c \uc774\uc6a9\ud558\uc5ec \uae30\uc800 \\(v_0\\)\uc5d0\uc11c \\(p(T)\\)\ub97c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[\\begin{align}
    \np(T)(v_0)
    \n&= T^n (v_0 ) – \\alpha_{n-1} T^{n-1} (v_0) – \\cdots – \\alpha_1 T(v_0) – \\alpha_0 v_0 \\\\[5pt]
    \n&= T(v_{n-1}) – \\alpha_{n-1} v_{n-1} – \\cdots – \\alpha_1 v_1 – \\alpha_0 v_0 \\\\[5pt]
    \n&= \\mathbf{0}.
    \n\\end{align}\\]
    \n\ub2e4\uc74c\uc73c\ub85c \ub2e4\ub978 \uae30\uc800\uc6d0\uc18c \\(v_j,\\) \\(j > 0\\)\uc5d0\uc11c \\(p(T)\\)\ub97c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[\\begin{align}
    \np(T)(v_j)
    \n&= p(T)(T^j (v_0 )) \\\\[5pt]
    \n&= [ p(T) T^j ] (v_0 ) \\\\[5pt]
    \n&= [ T^j p(T) ] (v_0) \\\\[5pt]
    \n&= T^j ( p(T)(v_0)) \\\\[5pt]
    \n&= T^j (0) = \\mathbf{0}.
    \n\\end{align}\\]
    \n\uacc4\uc0b0 \uacfc\uc815\uc5d0\uc11c \ubcc0\ud658 \\(p(T)\\)\uc640 \\(T^j\\)\uac00 \uc11c\ub85c \uad50\ud658\ub420 \uc218 \uc788\ub294 \uc774\uc720\ub294 \\(\\Hom(V,\\,V)\\)\uac00 \\(K\\)-\ub300\uc218\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \uc774\ub85c\uc368 \\(p(T)\\)\uac00 \\(v_0\\)\ub97c \\(\\mathbf{0}\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ubbc0\ub85c \\(p(T)\\)\ub294 \ubaa8\ub4e0 \uae30\uc800\uc6d0\uc18c \\(v_0,\\) \\(v_1,\\) \\(\\cdots,\\) \\(v_{n-1}\\)\ub97c \\(\\mathbf{0}\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(p(T)\\)\ub294 \ud56d\uc0c1 \ud568\uc22b\uac12\uc774 \\(\\mathbf{0}\\)\uc778 \ubcc0\ud658\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    Cayley-Hamilton \uc815\ub9ac\uc758 \uc99d\uba85<\/h2>\n

    \uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ubcf4\uc870\uc815\ub9ac\ub97c \ubc14\ud0d5\uc73c\ub85c Cayley-Hamilton \uc815\ub9ac(\uc815\ub9ac 1)\ub97c \uc99d\uba85\ud558\uc790.<\/p>\n

    \\(n\\)\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790.<\/p>\n

    \\(n=1\\)\uc778 \uacbd\uc6b0\ub294 \uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ubcf4\uc870\uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74 \ucc28\uc6d0\uc774 \\(1\\)\uc778 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc740 \\(V\\) \uc790\uc2e0\uc774\uac70\ub098 \uc790\uba85\ud55c \ubd80\ubd84\uacf5\uac04 \ubfd0\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

    \uc774\uc81c \\(n > 1\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \ub9cc\uc57d \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04 \uc911 \\(T\\)-\ubd88\ubcc0\uc778 \uac83\uc774 \\(V\\) \uc790\uc2e0\uacfc \uc790\uba85\ud55c \ubd80\ubd84\uacf5\uac04 \ubfd0\uc774\ub77c\uba74 \ubcf4\uc870\uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc6d0\ud558\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(V\\)\uc758 \ubd80\ubd84\uacf5\uac04 \\(W_1\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(T\\)-\ubd88\ubcc0\uc774\uba74\uc11c \\(V\\) \uc790\uc2e0\uc774 \uc544\ub2c8\uace0 \uc790\uba85\ud55c \ubd80\ubd84\uacf5\uac04\ub3c4 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790.<\/p>\n

    \\(W_1\\)\uc758 \uae30\uc800\ub97c \\(B_1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(B_1\\)\uc744 \ud655\uc7a5\ud558\uc5ec \\(V\\)\uc758 \uae30\uc800 \\(B\\)\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \\(B\\)\uc758 \uae30\uc800\uc6d0\uc18c \uc911 \\(B_1\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uac83\ub4e4\uc758 \ubaa8\uc784\uc744 \\(B_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(B_2\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \uacf5\uac04\uc744 \\(W_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V = W_1 \\oplus W_2\\)\uc774\ub2e4. <\/p>\n

    \uae30\uc800 \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc744 \\(A\\)\ub77c\uace0 \ud558\uc790. \\(T\\)\uac00 \\(W_1\\)\uc758 \uc6d0\uc18c\ub97c \\(W_1\\)\uc758 \uc6d0\uc18c\uc5d0 \ub300\uc751\uc2dc\ud0a4\ubbc0\ub85c \\(A\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\uc758 \ud589\ub82c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
    \n\\[A= \\left( \\begin{array}{c|c}
    \nP&Q \\\\
    \n\\hline
    \nO&R
    \n\\end{array} \\right).\\]
    \n\uc5ec\uae30\uc11c \\(P\\)\uc640 \\(R\\)\ub294 \uadf8 \ud06c\uae30\uac00 \uac01\uac01 \\(\\dim(W_1),\\) \\(\\dim(W_2)\\)\uc778 \uc815\uc0ac\uac01\ud589\ub82c\uc774\ub2e4. \\(W_2\\)\ub294 \\(T\\)-\ubd88\ubcc0\uc774 \uc544\ub2d0 \uc218\ub3c4 \uc788\uc73c\ubbc0\ub85c \ubd80\ubd84\ud589\ub82c \\(Q\\)\uc758 \uc131\ubd84 \uc911\uc5d0\ub294 \\(0\\)\uc774 \uc544\ub2cc \uac83\uc774 \uc874\uc7ac\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \\(T_R\\)\uac00 \uae30\uc800 \\(B_2\\)\uc5d0 \ub300\ud55c \ud45c\ud604\ud589\ub82c \\(B\\)\ub97c \uc67c\ucabd\uc5d0 \uacf1\ud558\ub294 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(T_R\\)\ub294 \\(W_2\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \uc774\uc81c \\(T\\)\uc640 \\(T_R\\) \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \ubc1d\ud600\uc57c \ud55c\ub2e4. \ud2b9\uc131\ub2e4\ud56d\uc2dd \\(p(t)\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4.
    \n\\[p(t) = p_1 (t) p_2 (t) \\]
    \n\uc5ec\uae30\uc11c \\(p_1(t)\\)\ub294 \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(W_1\\)\uc73c\ub85c \ucd95\uc18c\ud55c \\(T \\vert _{W_1}\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774\uba70, \\(p_2 (t)\\)\ub294 \\(T_R\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774\ub2e4. \\(A\\)\uc758 \uad6c\uc870\uc801 \ud2b9\uc131\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec \\(T_R(w)\\)\ub294 \\(W_1\\)\uc758 \uc6d0\uc18c \ub9cc\ud07c \ucc28\uc774\ub09c\ub2e4. \uc989 \uc784\uc758\uc758 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[T(w) – T_R (w) \\in W_1\\]
    \n\uc774\ub2e4. [\uc5ec\uae30\uc11c \uc8fc\ubaa9\ud560 \uc810\uc740 \\(T_R\\)\uac00 \\(B_2\\)\uc758 \uc6d0\uc18c\uc5d0 \uc791\uc6a9\ud558\uba74 \\(B_2\\)\uc758 \uc6d0\uc18c\uc758 \uc77c\ucc28\uacb0\ud569\uc740 \ud589\ub82c \\(R\\)\uc5d0 \uc758\ud574 \uc815\ud574\uc9c4\ub2e4\ub294 \uc810\uc774\ub2e4. \uadf8\ub7ec\uba74 \uadf8 \uc77c\ucc28\uacb0\ud569\uc5d0 \\(Q\\)\uc5d0 \uc758\ud574 \uc815\ud574\uc9c0\ub294 \\(B_1\\)\uc758 \uc6d0\uc18c\uc758 \uc77c\ucc28\uacb0\ud569\uc744 \ub354\ud55c\ub2e4.]<\/p>\n

    \uc591\ubcc0\uc5d0 \\(T\\)\ub97c \ud55c \ubc88 \ub354 \ucde8\ud558\uace0 \\(T\\)\uac00 \\(W_1\\)-\ubd88\ubcc0\uc774\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uba74 \uc784\uc758\uc758 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
    \n\\[T^2 (w) – TT_R (w) \\in W_1.\\]
    \n\uc88c\ubcc0\uc758 \ub450 \ubc88\uc9f8 \ud56d\uc5d0\uc11c \\(T\\)\ub97c \\(T_R\\)\ub85c \ubc14\uafb8\uc5b4\ub3c4 \ub3d9\uc77c\ud558\ubbc0\ub85c \uc784\uc758\uc758 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
    \n\\[T^2 (w) – T_R ^2 (w) \\in W_1 .\\]
    \n\uc774 \uacfc\uc815\uc744 \uacc4\uc18d \ubc18\ubcf5\ud558\uba74 \uc74c\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc815\uc218 \\(j\\)\uc640 \uc784\uc758\uc758 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[T^j (w) – T_R^j (w) \\in W_1\\]
    \n\uc774 \uc131\ub9bd\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc54c \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \ub2e4\ud56d\uc2dd \\(q(t) \\in K[t]\\)\uc640 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
    \n\\[q(T) (w) – q(T_R ) (w) \\in W_1 .\\]
    \n\uc774\uc81c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1 \\(p(T) = p_1 (T) p_2 (T) = p_2 (T) p_1 (T)\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc774 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc774 \\(W_1\\)\uc758 \uc6d0\uc18c\uc5d0 \uc791\uc6a9\ud558\ub294 \uacb0\uacfc\ub97c \uc0b4\ud3b4\ubcf4\uae30 \uc704\ud574 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc790. \\(p_1 (T)\\)\ub294 \\(W_1\\)\uc5d0\uc11c \ubaa8\ub4e0 \uac12\uc744 \\(\\mathbf{0}\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\uba70, \ub354\uc6b1\uc774 \uc784\uc758\uc758 \\(w\\in W_1\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
    \n\\[p(T) (w) = [p_2 (T) p_1 (T)] (w) = p_2 (T)(\\mathbf{0}) = \\mathbf{0}.\\]
    \n\ub2e4\uc74c\uc73c\ub85c \\(p(T)\\)\uac00 \\(W_2\\)\uc758 \uc6d0\uc18c\uc5d0 \uc791\uc6a9\ud558\ub294 \uacb0\uacfc\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc55e\uc11c \uc0b4\ud3b4\ubcf8 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[p_2 (T) (w) – p_2 (T_R ) (w) \\in W_1\\]
    \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(p_2 (T_R)\\)\ub294 \\(W_2\\)\uc758 \ubaa8\ub4e0 \uac12\uc744 \\(\\mathbf{0}\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\ubbc0\ub85c, \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec \\(p_2 (T)(w) \\in W_1\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \\(w\\in W_2\\)\uc5d0 \ub300\ud558\uc5ec \\(w_1 \\in W\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(p_2 (T) (w) = w_1\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba70,
    \n\\[p(T) (w) = [p_1 (T) p_2 (T) ] (w) = p_1 (T)(w_1) = \\mathbf{0}\\]
    \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \\(p(T)\\)\ub294 \\(W_1\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc640 \\(W_2\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub97c \\(\\mathbf{0}\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(p(T)\\)\ub294 \\(V\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub97c \\(\\mathbf{0}\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4.<\/p>\n

    \uc774\ub85c\uc368 \uc815\ub9ac 1\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/span><\/p>\n

    <\/p>\n

    \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \uc0bc\uac01\ud589\ub82c \ud45c\ud604<\/h2>\n

    \\(T\\)\uac00 \ubca1\ud130\uacf5\uac04 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(V\\)\uc758 \uc801\ub2f9\ud55c \uae30\uc800 \\(B\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \uc704\uc0bc\uac01\ud589\ub82c\uc774 \ub418\uba74 \u2018\\(T\\)\ub294 \uc0bc\uac01\ud589\ub82c\ub85c \ucd95\uc57d\ub420 \uc218 \uc788\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \\(T\\)\ub97c \uc0bc\uac01\ud589\ub82c\ub85c \ud45c\ud604\ud558\uba74 \\(T\\)\uc758 \uace0\uc733\uac12\uc744 \uc27d\uac8c \ucc3e\uc744 \uc218 \uc788\uc73c\ubbc0\ub85c \uc720\uc6a9\ud558\ub2e4. \ub354\uc6b1\uc774 \ub9cc\uc57d \\(T\\)\uac00 \uac00\uc5ed\uc774\uace0 \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \uc0bc\uac01\ud589\ub82c\uc774\uba74 \\(T\\)\uc758 \uc5ed\uc0ac\uc0c1\uc744 \uc27d\uac8c \ucc3e\uc744 \uc218 \uc788\ub2e4. \ud3b8\uc758\uc0c1 \uc9c0\uae08\ubd80\ud130 \uc0bc\uac01\ud589\ub82c\uc740 \uc704\uc0bc\uac01\ud589\ub82c\uc744 \uc774\ub974\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 4.<\/span>
    \n\\(V\\)\uac00 \uccb4 \\(K\\) \uc704\uc5d0\uc11c\uc758 \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T\\)\uac00 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub418\uba74 \\(T\\)\ub294 \uc0bc\uac01\ud589\ub82c\ub85c \ucd95\uc57d\ub420 \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n

    \n

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

    \\(n\\)\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc790. \uc99d\uba85 \uacfc\uc815\uc740 Cayley-Hamilton \uc815\ub9ac\uc758 \uc99d\uba85\uacfc \uc720\uc0ac\ud558\ub2e4.<\/p>\n

    \\(n=1\\)\uc778 \uacbd\uc6b0\ub294 \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(n > 1\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790.<\/p>\n

    \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \\(p(t)\\)\ub77c\uace0 \ud558\uace0 \\(p(t)=0\\)\uc758 \ud55c \uadfc\uc744 \\(\\lambda_1\\)\uc774\ub77c\uace0 \ud558\uc790. \uc815\ub9ac\uc758 \uac00\uc800\uc5e5 \uc758\ud558\uc5ec \\(\\lambda_1 \\in K\\)\uc774\uba70, \\(\\lambda_1\\)\uc740 \\(T\\)\uc758 \uace0\uc733\uac12\uc774\ub2e4. \\(\\lambda_1\\)\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\ubca1\ud130\ub97c \\(v_1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(v_1\\)\uc740 \\(V\\)\uc758 \\(1\\)\ucc28\uc6d0 \ubd80\ubd84\uacf5\uac04 \\(W_1\\)\uc744 \uc0dd\uc131\ud558\uba70, \\(W_1\\)\uc740 \\(T\\)-\ubd88\ubcc0\uc778 \uacf5\uac04\uc774 \ub41c\ub2e4. \\(v_1\\)\uc744 \ud655\uc7a5\ud558\uc5ec \\(V\\)\uc758 \uae30\uc800 \\(v_1,\\) \\(v_2,\\) \\(\\cdots,\\) \\(v_n\\)\uc744 \ub9cc\ub4e4\uc790. \uadf8\ub9ac\uace0 \\(v_2 ,\\) \\(\\cdots,\\) \\(v_n\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \uacf5\uac04\uc744 \\(W_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V = W_1 \\oplus W_2\\)\uc774\ub2e4. [\ubb3c\ub860 \\(W_2\\)\ub294 \\(T\\)-\ubd88\ubcc0\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4.]<\/p>\n

    \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\uc774\ub2e4.
    \n\\[A= \\left( \\begin{array}{c|c}
    \n\\lambda_1 & \\ast \\\\
    \n\\hline
    \n0 & R
    \n\\end{array} \\right).\\]
    \n\uc5ec\uae30\uc11c \\(R\\)\ub294 \\((n-1)\\times (n-1)\\) \ud589\ub82c\uc774\uba70 \ubcc4\ud45c\ub294 \ubaa8\ub4e0 \uc131\ubd84\uc774 \uc2a4\uce7c\ub77c\uc778 \uc5f4\ubca1\ud130\uc774\ub2e4. \uae30\uc800 \\(v_2 ,\\) \\(\\cdots ,\\) \\(v_n\\)\uc5d0 \ub300\ud558\uc5ec \\(R\\)\ub85c \ud45c\ud604\ub418\ub294 \uc120\ud615\ubcc0\ud658\uc744 \\(T_R\\)\ub77c\uace0 \ud558\uc790. \uc989 \\(T_R\\)\ub294 \\(W_2\\) \uc704\uc5d0\uc11c\uc758 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \uadf8\ub7ec\uba74 \\(T\\)\uc640 \\(T_R\\)\ub294 \ubaa8\ub450 \\(W_2\\)\uc5d0\uc11c \ub3d9\uc77c\ud55c \uc791\uc6a9\uc744 \ud558\ub294 \ubcc0\ud658\uc774\uba70, \\(W_1\\)\uc758 \uc6d0\uc18c\uc5d0 \ub300\ud574\uc11c\ub9cc \ucc28\uc774\uac00 \ub09c\ub2e4.<\/p>\n

    \ub2e4\uc74c\uc73c\ub85c \\(T_R\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd \\(q(t)\\)\ub97c \uc0dd\uac01\ud558\uc790. \ud589\ub82c \\(tI_n – A\\)\uc758 \uccab\uc9f8 \ud589\uc744 \uae30\uc900\uc73c\ub85c \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud558\uba74
    \n\\[p(t) = (t-\\lambda_1 )q(t)\\]
    \n\ub97c \uc5bb\ub294\ub2e4. \uc5ec\uae30\uc11c \\(q(t)=0\\)\uc758 \ubaa8\ub4e0 \uadfc\uc740 \\(p(t)=0\\)\uc758 \uadfc\uc774\ub2e4. \uc989 \\(q(t)=0\\)\uc758 \ubaa8\ub4e0 \uadfc\uc740 \\(K\\)\uc5d0 \uc18d\ud55c\ub2e4. \ub530\ub77c\uc11c \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(W_2\\)\uc758 \uae30\uc800 \\(v_2 ‘,\\) \\(\\cdots ,\\) \\(v_n ‘\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc774 \uae30\uc800\uc5d0 \ub300\ud55c \\(T_R\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \uc0bc\uac01\ud589\ub82c\uc774 \ub41c\ub2e4. \uc774 \uc0c8\ub85c\uc6b4 \ud45c\ud604\ud589\ub82c\uc744 \\(R ‘ \\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

    \\(v_1,\\) \\(v_2 ‘ ,\\) \\(\\cdots ,\\) \\(v_n ‘\\)\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uae30\uc800 \\(B ‘ \\)\uc740 \\(V\\)\uc758 \uae30\uc800\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(T\\)\uc640 \\(T_R\\)\ub294 \\(W_1\\)\uc758 \uc6d0\uc18c\uc5d0 \ub300\ud574\uc11c\ub9cc \ud568\uc22b\uac12\uc774 \ub2e4\ub974\ubbc0\ub85c \\(B ‘ \\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\uc774 \ub41c\ub2e4.
    \n\\[A ‘ = \\left( \\begin{array}{c|c}
    \n\\lambda_1 & \\ast \\\\
    \n\\hline
    \n0 & R ‘
    \n\\end{array} \\right).\\]
    \n\uc5ec\uae30\uc11c \\(R ‘ \\)\uc774 \uc0bc\uac01\ud589\ub82c\uc774\ubbc0\ub85c \\(A ‘ \\) \ub610\ud55c \uc0bc\uac01\ud589\ub82c\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \uc815\ub9ac 4\uc640 \uae30\uc800\uc758 \ubcc0\ud658\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \ub530\ub984\uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \ub530\ub984\uc815\ub9ac 5.<\/span>
    \n\\(A\\in M_n (K)\\)\uc774\uace0 \\(A\\)\uac00 \\(K\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud589\ub82c \\(P\\in \\operatorname{GL}_n (K)\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B = P^{-1} AP\\)\uac00 \uc0bc\uac01\ud589\ub82c\uc774 \ub41c\ub2e4.\n<\/p>\n<\/div>\n

    \ubcf5\uc18c\uc218 \uc9d1\ud569\uc740 \ub300\uc218\uc801\uc73c\ub85c \ub2eb\ud600 \uc788\uc73c\ubbc0\ub85c \uc815\ub9ac 4\ub85c\ubd80\ud130 \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \ub530\ub984\uc815\ub9ac 6.<\/span>
    \n\ubcf5\uc18c\ubca1\ud130\uacf5\uac04 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubaa8\ub4e0 \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc740 \uc0bc\uac01\ud589\ub82c\ub85c \ucd95\uc57d\ub420 \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n

    \ub0b4\uc801\uacf5\uac04\uc5d0 \uc815\ub9ac 4\ub97c \uc801\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 7.<\/span>
    \n\\(V\\)\uac00 \uc2e4\ub0b4\uc801\uacf5\uac04 \ub610\ub294 \ubcf5\uc18c\ub0b4\uc801\uacf5\uac04\uc774\uace0 \\(T\\)\uac00 \\(V\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc790\uae30\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(T\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(V\\)\uac00 \uc815\uc758\ub41c \uccb4\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc5d0\uc11c\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800 \\(B\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \uc0bc\uac01\ud589\ub82c\uc774 \ub41c\ub2e4.\n<\/p>\n<\/div>\n

    \n

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

    \uc99d\uba85 \uacfc\uc815\uc740 \uc815\ub9ac 4\uc758 \uc99d\uba85\uacfc \uac70\uc758 \uac19\ub2e4. \ub2e8, \uc911\uac04\uc5d0 \\(W_2 = W_1 ^\\bot\\)\uc744 \ud0dd\ud558\ub294 \ubd80\ubd84\ub9cc \ub2e4\ub974\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \uc9c0\uae08\uae4c\uc9c0 \ub17c\uc758\ud55c \ub0b4\uc6a9\uc744 \uc2e4\ud589\ub82c\uacfc \ubcf5\uc18c\ud589\ub82c\uc5d0 \uc801\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \ub530\ub984\uc815\ub9ac 8.<\/span>
    \n\\(A\\)\uac00 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uace0 \uadf8 \uc131\ubd84\uc774 \ubaa8\ub450 \uc2e4\uc218\uc774\uac70\ub098, \ubaa8\ub450 \ubcf5\uc18c\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(A\\)\uac00 \uc2e4\ud589\ub82c\uc774\uba74 \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc774 \\(\\mathbb{R}\\)\uc5d0\uc11c \uc77c\ucc28\uc2dd\uc758 \uacf1\uc73c\ub85c \uc778\uc218\ubd84\ud574\ub41c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc720\ub2c8\ud0c0\ub9ac \ud589\ub82c \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B = P^{-1} AP\\)\uac00 \uc0bc\uac01\ud589\ub82c\uc774 \ub41c\ub2e4.\n<\/p>\n<\/div>\n

    \n

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

    \\(\\mathbb{R}^n\\)\uacfc \\(\\mathbb{C}^n\\)\uc5d0\uc11c, \uc815\uaddc\uc9c1\uad50\ubca1\ud130\ub97c \ubcf4\ud1b5\uae30\uc800\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \uae30\uc800\ubcc0\ud658\ud589\ub82c \\(P\\)\ub294 \uc720\ub2c8\ud0c0\ub9ac \ud589\ub82c\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n


    \n<\/p>\n


    \n<\/p>\n

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

    \uc815\uc0ac\uac01\ud589\ub82c\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \uc774\uc6a9\ud55c \ud765\ubbf8\ub85c\uc6b4 \ub4f1\uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\(A\\)\uac00 \uc774\ucc28\uc7a5\uc0ac\uac01\ud589\ub82c\uc774\uace0 \\(A = \\left(\\begin{array}{cc} a & b \\\\ c & d \\end{array}\\right)\\) \uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(A^2 – (a+d)A + (ad-bc)I_2 = O.\\) \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc744 \\(p(t)\\)\ub77c\uace0 \ud558\uace0 \\(t=A\\)\ub97c \ub300\uc785\ud568\uc73c\ub85c\uc368 \uc704 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uac04\ub2e8\ud558\uac8c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \\(p(A) = O.\\) \uc774 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc740 \uc6b0\uc5f0\uc774 \uc544\ub2c8\uba70, Cayley-Hamilton \uc815\ub9ac\uc758 \uacb0\uacfc\uc774\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud2b9\uc131\ub2e4\ud56d\uc2dd\uc758 \uc131\uc9c8\uacfc \\(T\\)-\ubd88\ubcc0 \uacf5\uac04\uc758 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf4\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":[482,479,465,484,485,426,487,486,483,471,458],"class_list":["post-5721","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-cayley-hamilton","tag-t-","tag-465","tag-484","tag-485","tag-426","tag-487","tag-486","tag-483","tag-471","tag-458"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5721","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=5721"}],"version-history":[{"count":22,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5721\/revisions"}],"predecessor-version":[{"id":5745,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5721\/revisions\/5745"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5721"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5721"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5721"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}