{"id":5386,"date":"2020-10-05T23:34:13","date_gmt":"2020-10-05T14:34:13","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5386"},"modified":"2020-10-05T23:54:45","modified_gmt":"2020-10-05T14:54:45","slug":"calculus-cauchy-schwarz-inequality-by-orthogonal-decomposition","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-cauchy-schwarz-inequality-by-orthogonal-decomposition\/","title":{"rendered":"\ubca1\ud130\uc758 \uc9c1\uad50\ubd84\ud574\ub97c \uc774\uc6a9\ud55c \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd \uc99d\uba85"},"content":{"rendered":"

\ubca1\ud130\uc758 \uc9c1\uad50\ubd84\ud574\ub97c \uc774\uc6a9\ud558\uc5ec \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n

\\(V\\)\uac00 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(\\mathbf{u},\\,\\mathbf{v}\\in V\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc774 \\(\\mathbf{0}\\)\uc774\uba74 \uc790\uba85\ud558\uac8c
\n\\[ \\lvert \\langle \\mathbf{u} ,\\, \\mathbf{v} \\rangle \\rvert \\le \\lVert \\mathbf{u} \\rVert \\lVert \\mathbf{v} \\rVert \\tag{1}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\) \uc911 \uc5b4\ub290\uac83\ub3c4 \\(\\mathbf{0}\\)\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0
\n\\[\\mathbf{w} = \\mathbf{u} – \\frac{\\langle \\mathbf{u} ,\\, \\mathbf{v} \\rangle}{\\lVert \\mathbf{v} \\rVert^2} \\mathbf{v} \\tag{2}\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\mathbf{u}\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc11c\ub85c \uc218\uc9c1\uc778 \ub450 \ubca1\ud130\uc758 \ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4.
\n\\[\\mathbf{u} = \\frac{\\langle \\mathbf{u} ,\\, \\mathbf{v} \\rangle}{\\lVert \\mathbf{v} \\rVert ^2} \\mathbf{v} + \\mathbf{w} .\\tag{3}\\]
\n\ub530\ub77c\uc11c \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\n \\lVert \\mathbf{u} \\rVert^2
\n&= \\left\\lVert \\frac{\\langle \\mathbf{u} ,\\, \\mathbf{v} \\rangle}{\\lVert \\mathbf{v} \\rVert^2} \\mathbf{v} \\right\\rVert^2 + \\lVert \\mathbf{w} \\rVert^2 \\\\[4pt]
\n&= \\frac{\\lvert \\langle \\mathbf{u} ,\\, \\mathbf{v} \\rangle \\rvert^2}{\\lVert\\mathbf{v}\\rVert^2} + \\lVert\\mathbf{w}\\rVert^2 \\\\[4pt]
\n&\\ge \\frac{ \\lvert\\langle\\mathbf{u},\\,\\mathbf{v}\\rangle\\rvert^2 }{\\lVert\\mathbf{v}\\rVert^2}.
\n\\tag{4}
\n\\end{align} \\]
\n\uc774 \ubd80\ub4f1\uc2dd\uc5d0 \\(\\lVert \\mathbf{v} \\rVert^2\\)\uc744 \uacf1\ud558\uace0 \uc591\uc758 \uc81c\uacf1\uadfc\uc744 \ucde8\ud558\uba74 (1)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c (1)\uc5d0\uc11c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub294 \uc870\uac74\uc744 \uad6c\ud558\uc790. (4)\ub97c \ubcf4\uba74 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc740 \\(\\mathbf{w} = \\mathbf{0}\\)\uc77c \ub54c\uc784\uc744 \uc54c \uc218 \uc788\ub2e4. \uc774\uac83\uc740 (2)\uc758 \uc6b0\ubcc0\uc774 \\(\\mathbf{0}\\)\uc784\uc744 \uc758\ubbf8\ud558\ubbc0\ub85c, \\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\)\uac00 \ud3c9\ud589\ud560 \ub54c (1)\uc5d0\uc11c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

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

\ubca1\ud130\uc758 \uc9c1\uad50\ubd84\ud574\ub97c \uc774\uc6a9\ud558\uc5ec \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \\(V\\)\uac00 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(\\mathbf{u},\\,\\mathbf{v}\\in V\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc774 \\(\\mathbf{0}\\)\uc774\uba74 \uc790\uba85\ud558\uac8c \\( \\lvert \\langle \\mathbf{u} ,\\, \\mathbf{v} \\rangle \\rvert \\le \\lVert \\mathbf{u} \\rVert \\lVert \\mathbf{v} \\rVert \\) \ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\) \uc911 \uc5b4\ub290\uac83\ub3c4 \\(\\mathbf{0}\\)\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(\\mathbf{w} = \\mathbf{u} – \\frac{\\langle \\mathbf{u} ,\\, \\mathbf{v} \\rangle}{\\lVert \\mathbf{v} \\rVert^2} \\mathbf{v} \\) \ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\mathbf{u}\\)\ub294…<\/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":[424,427,400,399,426,425],"class_list":["post-5386","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-cauchy-schwarz","tag-inequality","tag-linear-algebra","tag-399","tag-426","tag-425"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5386","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=5386"}],"version-history":[{"count":13,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5386\/revisions"}],"predecessor-version":[{"id":5399,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5386\/revisions\/5399"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5386"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5386"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}