{"id":5358,"date":"2020-10-04T22:12:35","date_gmt":"2020-10-04T13:12:35","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5358"},"modified":"2020-10-05T23:33:12","modified_gmt":"2020-10-05T14:33:12","slug":"calculus-cauchy-schwarz-inequality-by-lagrangian","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-cauchy-schwarz-inequality-by-lagrangian\/","title":{"rendered":"\ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95\uc744 \uc774\uc6a9\ud55c \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd \uc99d\uba85"},"content":{"rendered":"

\\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc774\uace0 \\(a_1 ,\\) \\(\\cdots ,\\) \\(a_n ,\\) \\(b_1 ,\\) \\(\\cdots,\\) \\(b_n\\)\uc774 \ubaa8\ub450 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\left( \\sum_{i=1}^n a_i b_i \\right)^2 \\le \\left( \\sum_{i=1}^n a_i ^2\\right) \\left(\\sum_{i=1}^n b_i ^2 \\right).\\tag{1}\\]
\n\uc774 \ubd80\ub4f1\uc2dd\uc744 \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd(Cauchy-Schwarz inequality)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95(method of Lagrange’s multiplier)\uc744 \uc774\uc6a9\ud558\uc5ec \uc774 \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \uc99d\uba85\uc744 \ub9c8\uce60 \ub54c\uae4c\uc9c0 \ucca8\uc218 \\(i\\)\uc640 \\(j\\)\ub294 \\(n\\) \uc774\ud558\uc758 \uc790\uc5f0\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n

\uc99d\uba85 \uacfc\uc815\uc740 \ub450 \ub2e8\uacc4\ub85c \uc9c4\ud589\ub41c\ub2e4. \uccab\uc9f8 \ub2e8\uacc4\uc5d0\uc11c\ub294 \uae38\uc774\uac00 \\(1\\)\uc778 \ub450 \ubca1\ud130 \\(\\mathbb{x}=(x_i)\\)\uc640 \\(\\mathbb{y}=(y_i)\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[-1 \\le \\mathbb{x}\\cdot\\mathbb{y} \\le 1 \\]
\n\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc778\ub2e4. \ub458\uc9f8 \ub2e8\uacc4\uc5d0\uc11c\ub294 \\(\\mathbb{a} = (a_i ),\\) \\(\\mathbb{b} = (b_i )\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\mathbb{x} = \\frac{\\mathbb{a}}{\\lVert \\mathbb{a} \\rVert} , \\,\\,
\n\\mathbb{y} = \\frac{\\mathbb{b}}{\\lVert \\mathbb{b} \\rVert}
\n \\]
\n\ub77c\uace0 \ub450\uace0 (1)\uc744 \ub04c\uc5b4\ub0b8\ub2e4.<\/p>\n

1\ub2e8\uacc4.<\/h4>\n

\ud568\uc218 \\(f,\\) \\(g,\\) \\(h\\)\ub97c \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[\\begin{align}
\nf(x_1 ,\\, \\cdots ,\\, x_n ,\\, y_1 ,\\, \\cdots,\\, y_n ) &= \\sum_{i=1}^n x_i\\, y_i , \\\\[5pt]
\ng(x_1 ,\\, \\cdots ,\\, x_n ,\\, y_1 ,\\, \\cdots,\\, y_n ) &= \\sum_{i=1}^n x_i ^2 ,\\\\[5pt]
\nh(x_1 ,\\, \\cdots ,\\, x_n ,\\, y_1 ,\\, \\cdots,\\, y_n ) &= \\sum_{i=1}^n y_i^2 .
\n\\end{align}\\]
\n\uc774 \uc815\uc758\uc5d0 \uc758\ud558\uba74 \ud568\uc218 \\(f\\)\ub294 \\(2n\\)\uac1c\uc758 \ubcc0\uc218\ub97c \uac00\uc9c0\uace0 \uc788\uace0, \ud568\uc218 \\(g,\\) \\(h\\)\ub294 \\(n\\)\uac1c\uc758 \ubcc0\uc218\ub97c \uac00\uc9c4 \uc148\uc774\ub2e4. \uc138 \ud568\uc218\uc758 \uae30\uc6b8\uae30(gradient)\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\n\\nabla f &= (y_1 ,\\, \\cdots ,\\, y_n ,\\, x_1 ,\\, \\cdots ,\\, x_n ), \\\\[8pt]
\n\\nabla g &= ( 2x_1 ,\\, \\cdots ,\\, 2x_n ,\\, 0 ,\\, \\cdots ,\\, 0 ), \\\\[8pt]
\n\\nabla h &= ( 0,\\, \\cdots ,\\, 0 ,\\, 2y_1 ,\\, \\cdots ,\\, 2y_n ).
\n\\end{align}\\]
\n\uc774\uc81c \uc801\ub2f9\ud55c \uc0c1\uc218 \\(\\lambda,\\) \\(\\mu\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790.
\n\\[\\nabla f = \\lambda \\nabla g + \\mu \\nabla h .\\tag{2}\\]
\n\uc774 \ubc29\uc815\uc2dd\uc744 \\(\\lambda\\)\uc5d0 \ub300\ud558\uc5ec \ud480\uc790. \\(1\\le i \\le n\\)\uc778 \\(i\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[y_i = 2\\lambda x_i ,\\,\\, x_i = 2\\mu y_i \\tag{3}\\]
\n\uc774\ub2e4. \ub9cc\uc57d
\n\\[g=1,\\,\\,h=1 \\tag{4}\\]
\n\uc774\ub77c\ub294 \uc81c\ud55c \uc870\uac74\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \uac00\uc815\ud558\uba74 (3)\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[1=\\sum_{i=1}^n y_i^2 = \\sum_{i=1}^n 4 \\lambda^2 x_i^2 = 4 \\lambda^2 \\sum _{i=1} ^n x_i^2 = 4\\lambda^2 .\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(\\lambda = \\pm 1\/2\\)\uc774\ub2e4.<\/p>\n

\ub9cc\uc57d \\(\\lambda = 1\/2\\)\uc774\uba74, (3)\uc5d0 \uc758\ud558\uc5ec \\(1\\le i \\le n\\)\uc778 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(y_i = x_i\\)\uc774\ubbc0\ub85c
\n\\[\\sum_{i=1}^n x_i \\,y_i = \\sum_{i=1}^n x_i^2 = 1\\tag{5}\\]
\n\uc774\ub2e4. \ub9cc\uc57d \\(\\lambda = -1\/2\\)\uc774\uba74, (3)\uc5d0 \uc758\ud558\uc5ec \\(1\\le i \\le n\\)\uc778 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(y_i = -x_i\\)\uc774\ubbc0\ub85c
\n\\[\\sum_{i=1}^n x_i \\,y_i = -\\sum_{i=1}^n x_i^2 = -1\\tag{6}\\]
\n\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \ubc29\uc815\uc2dd (2)\ub97c \\(\\mu\\)\uc5d0 \ub300\ud558\uc5ec \ud480\uc790. \uadf8\ub7ec\uba74 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(\\mu = \\pm 1\/2\\)\uc744 \uc5bb\uc73c\uba70, \ubaa8\ub4e0 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(y_i = \\pm x_i\\)\uc640
\n\\[\\sum_{i=1}^n x_i \\,y_i = \\pm 1\\]
\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(g=1,\\) \\(h=1\\)\uc774\ub77c\ub294 \uc81c\ud55c \uc870\uac74 \uc544\ub798\uc5d0\uc11c
\n\\[f=\\sum_{i=1}^n x_i\\,y_i\\]
\n\uc758 \uac12\uc758 \ucd5c\uc19f\uac12\uc740 \\(-1\\)\uc774\uace0, \ucd5c\ub313\uac12\uc740 \\(1\\)\uc774\ub2e4.<\/p>\n

2\ub2e8\uacc4.<\/h4>\n

\uc774\uc81c \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud560 \uc900\ube44\uac00 \ub418\uc5c8\ub2e4. \ud45c\uae30\ub97c \ud3b8\ud558\uac8c \ud558\uae30 \uc704\ud558\uc5ec
\n\\[A = \\sum_{j=1}^n a_j^2 ,\\,\\, B = \\sum_{j=1}^n b_j^2\\]
\n\uc774\ub77c\uace0 \uc815\uc758\ud558\uc790. \ub9cc\uc57d \\(A=0\\) \ub610\ub294 \\(B=0\\)\uc774\ub77c\uba74 \uc790\uba85\ud558\uac8c \uc6d0\ud558\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A \\ne 0,\\) \\(B \\ne 0\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uace0 \uc99d\uba85\ud558\uc790. \\(1\\le i\\le n\\)\uc778 \\(i\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[x_i = \\frac{a_i}{\\sqrt{A}} ,\\,\\, y_i = \\frac{b_i}{\\sqrt{B}}\\]
\n\ub77c\uace0 \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\sum_{i=1}^n x_i^2 = 1 ,\\,\\, \\sum_{i=1}^n y_i^2 = 1\\]
\n\uc774\ubbc0\ub85c (4)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370
\n\\[\\sum_{i=1}^n \\frac{a_i \\,b_i}{\\sqrt{A} \\sqrt{B}} = \\sum_{i=1}^n x_i\\,y_i\\]
\n\uc774\ubbc0\ub85c, \uc55e\uc758 \ub17c\uc758 \uacfc\uc815\uc5d0 \uc758\ud558\uc5ec \uc774 \uac12\uc758 \ubc94\uc704\ub294 \\(-1\\) \uc774\uc0c1 \\(1\\) \uc774\ud558\uc774\ub2e4. \uc989
\n\\[-1 \\le \\sum_{i=1}^n \\frac{a_i \\,b_i}{\\sqrt{A} \\sqrt{B}} \\le 1 \\]
\n\uc774\ub2e4. \\(\\sqrt{A} \\sqrt{B}\\)\ub97c \uacf1\ud558\uba74
\n\\[-\\sqrt{A} \\sqrt{B} \\le \\sum_{i=1}^n a_i \\, b_i \\le \\sqrt{A} \\sqrt{B}\\]
\n\uc774\ubbc0\ub85c (1)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\ud55c\ud3b8 (5)\uc640 (6)\uc5d0 \uc758\ud558\uc5ec, (1)\uc5d0\uc11c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc740 \ubaa8\ub4e0 \\(i\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\frac{a_i}{\\sqrt{A}} = \\frac{b_i}{\\sqrt{B}}\\]
\n\ub610\ub294
\n\\[\\frac{a_i}{\\sqrt{A}} = -\\frac{b_i}{\\sqrt{B}}\\]
\n\uc77c \ub54c, \uc989 \ub450 \ubca1\ud130 \\((a_1 ,\\, \\cdots ,\\, a_n)\\)\uacfc \\((b_1 ,\\, \\cdots ,\\, b_n )\\)\uc774 \ud3c9\ud589\ud560 \ub54c\uc774\ub2e4.<\/p>\n

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

\\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc774\uace0 \\(a_1 ,\\) \\(\\cdots ,\\) \\(a_n ,\\) \\(b_1 ,\\) \\(\\cdots,\\) \\(b_n\\)\uc774 \ubaa8\ub450 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\left( \\sum_{i=1}^n a_i b_i \\right)^2 \\le \\left( \\sum_{i=1}^n a_i ^2\\right) \\left(\\sum_{i=1}^n b_i ^2 \\right).\\) \uc774 \ubd80\ub4f1\uc2dd\uc744 \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd(Cauchy-Schwarz inequality)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95(method of Lagrange’s multiplier)\uc744 \uc774\uc6a9\ud558\uc5ec \uc774 \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \uc99d\uba85\uc744 \ub9c8\uce60 \ub54c\uae4c\uc9c0 \ucca8\uc218 \\(i\\)\uc640 \\(j\\)\ub294 \\(n\\) \uc774\ud558\uc758 \uc790\uc5f0\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \uc99d\uba85 \uacfc\uc815\uc740 \ub450…<\/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":[47],"tags":[424,386,396,399,425],"class_list":["post-5358","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-cauchy-schwarz","tag-lagrangian-method","tag-396","tag-399","tag-425"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5358","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=5358"}],"version-history":[{"count":27,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5358\/revisions"}],"predecessor-version":[{"id":5385,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5358\/revisions\/5385"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}