From a4b945b2c52021eac2935e7445cd6f7a37402348 Mon Sep 17 00:00:00 2001 From: songyc macbook Date: Wed, 12 Aug 2026 00:55:57 +0900 Subject: [PATCH] =?UTF-8?q?4.2=20=EC=A7=81=EA=B5=90=EC=82=AC=EC=98=81=20?= =?UTF-8?q?=EC=A1=B0=EA=B1=B4(P=E1=B5=80=3DP)=20=EB=B3=B4=EC=B6=A9:=20?= =?UTF-8?q?=EB=B9=84=EC=8A=A4=EB=93=AC=ED=95=9C=20=EC=82=AC=EC=98=81=20?= =?UTF-8?q?=EB=B0=98=EB=A1=80=20+=20=EA=B8=B0=EB=B3=B8=EC=A0=95=EB=A6=AC?= =?UTF-8?q?=202=20=EA=B8=B0=EB=B0=98=20=EC=84=A4=EB=AA=85=20+=20=EC=9E=90?= =?UTF-8?q?=EC=9E=91=20=EA=B7=B8=EB=A6=BC?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01Ufr6wUhJAzBaqcaBiRJ1xk --- .../4장_4.2_직교사영_vs_비스듬한사영.png | Bin 0 -> 118043 bytes .../4장_4.2_직교사영_vs_비스듬한사영.svg | 46 ++++++++++++++++++ .../pages/4.2 직선과 부분공간 위로의 사영.md | 15 +++++- 3 files changed, 60 insertions(+), 1 deletion(-) create mode 100755 Major Study/assets/4장_4.2_직교사영_vs_비스듬한사영.png create mode 100755 Major Study/assets/4장_4.2_직교사영_vs_비스듬한사영.svg diff --git a/Major Study/assets/4장_4.2_직교사영_vs_비스듬한사영.png b/Major Study/assets/4장_4.2_직교사영_vs_비스듬한사영.png new file mode 100755 index 0000000000000000000000000000000000000000..32eaa436e4e03f0b5a349fbf8cc2fcc94b915084 GIT binary patch literal 118043 zcmeFZXIPU>+CPkn6&r$}bU{F=q6pG0NQX!hsgX`-p@*7?9i)hqD4{7$KtSm|7D5$} zgx*3Ap@mQqAPM;gy!YN+clZ7Jet9?!j!fp7Ysxw2ob@+}xM!ew;uzO4CMKp6+FI&H zOiTw?n3$MPA34ORDLfi{i}Atid`qA4`6ZTxYI~6JUBF(;NS}!*ScHk`(NiX-ZAR6j zIVL9mD@;uD51E)0-!d_AdZpADDl;CaJ3i2M(${AaW0a3D?Rx=YVrG=~G5)z2|4d8= zGWIbYV0`Y|t;^W|kEaJ$G7kKs%zS#c;bXJT8Ah{okjVqT2l{%7_FxYwTL-Y6qg0@W z*KPwQM7(ZkWtRv^&B-P2bwP(^UBg(9Q8TP-apu-C-TO-1m5{yl*^U>`>TIjJjB zR|Hj$2?z)%`#3l$8mVji^*ZC5ilB?1pO>PvGz0>Xf?SgV`#4L>C@3gMU%4uM^{OPJ zg`{tgr=M-0q^GaY?@s>FkGi9;y${IC4+QoU*zMQW4(#uzA}F{!(BFT*=jj*-`sYZV zzJD!?u|VnFC(<%fSET>$oAIjhZmr@yP@tpx19gyxqo*%p3{{!ySFb4Vz3?B8{yF5o zylVcxU;URyf4!;wl+)NrX?h1off$Hh-u$`<|BU=P6%Joxl^uT{~r(j zY8syMh^gV!{{QwWqlL@Oefy|~7TNU$|3~-xH&qY(=b(Ev8T$@0uW)}hmW}wY6JtJd zWSnyR&*}bSR)^S6e+7Rv=)U-0mbBYO;1S{4|Ckap*S(ue$bCw8$!Gt^(*LBzZ*g<{=09rYB*WJ_krFC9)7WFV{Qfwjf(Ld(pRznZh zWPz*pJ$QnSyF-%vR5|%w^)OGI>h|2!_v{X(E&{sG(%tF-0soA1huG!UhR1OjV2Xjr zGJ=g#E+{9h031`euJIVqOsfI6FGP*7NB)hZ`m&bmomlSYr%pSqmqUU>b! 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/dev/null +++ b/Major Study/assets/4장_4.2_직교사영_vs_비스듬한사영.svg @@ -0,0 +1,46 @@ + + + + + + + + + + + + + + P² = P인 두 "사영": 직교사영 vs 비스듬한 사영 + + + + 직선 L + + + + e = b − p, L과 직교 ✗ + + + + e* ⊥ L ✓ + + + + + b + + + + O + + + p* = 수선의 발 + (직교사영, Pᵀ = P: 가장 가까움) + p = Pb = (1, 1) + (비스듬한 사영: 옆으로 미끄러짐) + + + 둘 다 "두 번 사영하면 그대로" (P² = P): 이미 L 위의 점은 안 움직인다. + 하지만 가장 가까운 점(수선의 발)을 주는 것은 Pᵀ = P인 직교사영뿐! + diff --git a/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md b/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md index 962559b..242776e 100755 --- a/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md +++ b/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md @@ -69,7 +69,20 @@ deck:: Logseq/전공공부/선형대수학 - **사영행렬의 두 성질**: - $P^2 = P$: 두 번 사영해도 그대로다 (이미 직선 위에 있는 $p$ 를 또 사영하면 자기 자신, [3단계] 특별한 경우 ①). - 📌 원문 (p.219): ((6a7a040f-bcd4-4e55-863a-6152ed91aa7f)) - - $P^{\mathrm{T}} = P$ (대칭). ⚠️ 연습문제 0 (한국판 추가 문제)의 교훈: $P^2 = P$ 만으로는 "사영"이고, 오차 $e$ 가 $p$ 와 직교하는 **직교사영**이 되려면 $P^{\mathrm{T}} = P$ 까지 필요하다. 이 절의 $P$ 는 둘 다 만족한다. + - $P^{\mathrm{T}} = P$ (대칭): 이 절에서 만든 $P$ 는 자동으로 대칭이다 (연습문제 23: 전치해보면 그대로 $P$). + - ⚠️ **"어차피 정의상** $e \perp p$ **아닌가?"** (연습문제 0이 다루는 미묘한 지점): "수선의 발을 내린다"로 사영을 정의하면 직교는 당연하다. 그런데 대수 조건 $P^2 = P$ **만** 만족하는 행렬 중에는 수선의 발이 아닌 **비스듬한 사영**(oblique projection)도 있다. 그런 가짜(?)를 걸러내는 조건이 $P^{\mathrm{T}} = P$ 다. + - 사영의 넓은 정의는 "$P^2 = P$" 하나뿐이다: 결과가 $P$ 의 열공간에 떨어지고, 이미 거기 있는 벡터는 안 움직인다. "그림자의 그림자는 자기 자신"은 빛이 **비스듬히 들어와도** 성립한다. 즉 $P^2 = P$ 는 "L 위에 떨어뜨린다"까지만 보장하고, "수직으로 떨어뜨린다"는 보장하지 않는다. + - 🔢 반례 체험 (연습문제 0의 행렬): $P = \begin{bmatrix} 0&1 \\ 0&1 \end{bmatrix}$ 은 $P^2 = P$ 를 만족하고 직선 $y = x$ 위로 떨어뜨린다. 그런데 $b = (0, 1)$ 을 넣으면: + - $p = Pb = (1, 1)$, $e = b - p = (-1, 0)$, 내적 $p^{\mathrm{T}}e = -1 \neq 0$. **직교가 아니다!** + - 수선의 발은 $(\frac{1}{2}, \frac{1}{2})$ 인데, 이 $P$ 는 $b$ 를 $x$ 방향으로 **옆으로 미끄러뜨려** $(1,1)$ 에 떨어뜨린다. 거리도 $\|e\| = 1 > \frac{1}{\sqrt{2}}$ 라서 최단이 아니다. + - ![직교사영 vs 비스듬한 사영](../assets/4장_4.2_직교사영_vs_비스듬한사영.png) + - 정리하면: "정의상 이미 직교"라는 직관은 **수선의 발(직교사영)이라는 정의를 이미 채택했을 때** 이야기다. 이 절의 $P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$ 는 직교 조건에서 **유도**했으므로 자동으로 직교사영이 맞다. 문제는 방향이 반대일 때다: 정체 모를 행렬 $P$ 가 주어졌을 때 "이거 직교사영 맞아?"를 판정하려면 $P^2 = P$ (사영이긴 한가) 와 $P^{\mathrm{T}} = P$ (수선의 발인가) 를 **둘 다** 확인해야 한다. + - 왜 하필 "대칭"이 직교를 보장하나 (4.1 기본정리 2로 설명하면 명쾌하다): + - $p = Pb$ 는 열공간 $\mathbf{C}(P)$ 에 살고, $e = (I-P)b$ 는 영공간 $\mathbf{N}(P)$ 에 산다 ($Pe = Pb - P^2b = 0$). + - 그런데 기본정리 2가 보장하는 직교는 $\mathbf{N}(P) \perp \mathbf{C}(P^{\mathrm{T}})$, 즉 영공간 ⊥ **행공간**이다. 열공간과의 직교는 공짜가 아니다! + - $P^{\mathrm{T}} = P$ 면 행공간 = 열공간이 되므로 그제서야 $\mathbf{N}(P) \perp \mathbf{C}(P)$, 즉 $e \perp p$ 가 성립한다. + - 반례 행렬로 확인: 행공간은 $(0,1)$ 방향, 열공간은 $(1,1)$ 방향으로 서로 다르다. $e = (-1,0)$ 은 행공간과는 직교하지만 ✓ 열공간(사영이 떨어지는 곳)과는 직교하지 않는다 ✗. + - 대수 한 줄 확인: $p^{\mathrm{T}}e = (Pb)^{\mathrm{T}}(I-P)b = b^{\mathrm{T}}P^{\mathrm{T}}(I-P)b$ 인데, $P^{\mathrm{T}} = P$ 를 쓰면 $b^{\mathrm{T}}(P - P^2)b = 0$ (모든 $b$ 에서). - $I - P$ 도 사영행렬이다: - 📌 원문 (p.223): ((6a7af20e-41d5-4040-a603-a53e5809a41f)) / ((6a7af214-6b8f-4d35-99ab-549c781aa104)) - $Pb = p$ 이므로 $(I - P)b = b - p = e$. 즉 $I - P$ 는 **오차 쪽을 만들어주는 사영**이다.