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December 29, 2015

orthogonal matrix proof

/Differences[0/minus 54/negationslash] /Type/FontDescriptor 5@[B[^23$tdNcFij2CQr+"`iqX)m`KAI=T),"uu7OumGH*7muN^A2=81U-lK`]oq-#CJV#R*,4% stream /Descent -194 (Pji?8l.kqinn(Hf(Ga+RYT!59]c&f#h#RPC[#J(S0p\g:c+sQljO6S)]d7DHsXT /FontDescriptor 39 0 R 0%U9m"ptK0fcj+Xfh+\l]-nep#iA1CXKh1c4Mt;5RBH1@KfA?sqp%Ji4J. /FirstChar 12 >> >> trailer /Encoding 57 0 R X+QTFbKB3W11*.q#2<5O/!\DZGsIOO&^:Y-;tCQ+R8,-h;d,c?Gpq1=V!7K. endobj /Length 461 mathsfreak. /LastChar 110 56 0 obj endobj /Type/FontDescriptor /XHeight 697 Y.4W[%)$W?X)C&UGHo(^@H9VNY\gJ!3XG9R_bP4WI)(FkAmCC(,i.hU4_L)%BYmQK\%Zf9XM(oK/(X[MWf\: /BaseEncoding/MacRomanEncoding :0>4f6EBZcDi N62(k9bIB@ 26 0 obj /Type/Font endobj << /Length 4687 /Filter[/ASCII85Decode/FlateDecode] /Type/FontDescriptor /Differences[110/n] Proof. #JX$)lj[lhrb+!=2orsaWbqS;u#iWYHpAu'7LG*39O1L endobj /Flags 34 jlg"g1XLtWU1'85_SIZa6;@mXPYmJk+)ZIV7q%n]`q*!jZcRS$M%>-@,!T(oSt]8_WnHqA%VQZl endstream 0000047683 00000 n This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: Q T = Q − 1, {\displaystyle Q^{\mathrm {T} }=Q^{-1},} where Q − 1 {\displaystyle Q^{-1}} is the inverse of Q. 23 0 obj @ 9 0 obj Since det (A) = det (Aᵀ) and the determinant of product is the product of determinants when A is an orthogonal matrix. Let $\mathbf{u}, \mathbf{v}$ be eigenvectors corresponding to $\alpha, \beta$, respectively. /FirstChar 105 #8R9[1n$i_^Zigtp6gR4oTk@b(Q%tJeiY1[/VB,d&b[\"bsTbe2eG2=oh3&>*lXIS? >> @Xcgs3nHMi`(/^hBoK:Y0H@]T0^2-gsl;,"[qKIbH_IPF7hFPHG)=Ybs$$Ca])NaetRRMr0Oc2 /LastChar 48 /Widths[647] /LastChar 120 1C5B>Y%192'OUrTi2qiL8[EX_8'#Y.`$T^r^_%'? :,] << /Differences[11/ff 44/comma 46/period 61/equal 69/E 73/I 80/P/Q/R 84/T/U 97/a 99/c/d/e/f/g/h/i 108/l/m/n/o/p 114/r/s/t/u/v/w/x/y] 0000000015 00000 n (J%n`dMS(`a4?\Inl'ht#adC.:5>eHr? /StemV 114 12 0 obj /Subtype/Type1C `!Ts&0o$TAXmX,6Mfd3o7Q\%e^adR@X^7hS*FOjlQR4CAL?olA8[mn^,\'f#2YlS&9nJ$sA 16 0 obj endobj U6%Q_KZ%g&UguS'"2,4,-o!P!U]l7Z-PfQJCu+N8?+L6p/5fOSMF.X.W&SUB?V+)`p@_$:q#5;9'e+a,D8>iMVC4DfnkO$gQ_MXLUA\t1^36r*GMANr_[dBg@h /Subtype/Type1C /Subtype/Type1 /Widths[578 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 277 277 0 0 783 0 0 747 759 0 0 0 0 0 0 434 0 0 0 0 0 0 638 0 0 0 0 0 0 0 831 578 0 0 0 0 0 0 0 530 434 434 518 0 0 0 0 349 410 0 0 879 602 0 0 0 0 0 0 0 0 0 566 494] endobj /Subtype/Type1 endobj /Length1 454 /Parent 3 0 R ]HrkOaPAk>BpX,g0$b$(sT's:6dq(WgmjP_*\uP=e]?-A(QEoi_#LYAYD1Hr> /Length1 3333 `bdU,6XpCNJG?$qTKa/[5sP5coQtclZE=+rLZa5DkCJ/h! /Length1 345 /BaseEncoding/MacRomanEncoding )7*[l]k,kB4D&;`LZeT@ci)OoFL,tjZoQ>A&fI)#1&ZkB+9:rFHV^fJ])#GR,AuJHZF:%`p+mU, >> 0000026894 00000 n All the columns (v ₁, …, váµ¢, …) in Q are orthonormal, i.e. :+KKqlUrBQaaiIndFD!j7Jkq[L%7#390Ybf^HAl7,12NTXk This vector will run along $\vec{b}$. BB' = I. gC.Xsb&BMFhi+cLZ$Vj4HCGT4M#fu_9G]Hsi;sLmch$r=+:qYM.+V7,HZZCR8u8QS:J\((9=]"lNC7?p):>_R_IXSjO%%)geM[XG\oB /Length 429 Proof. 0000015737 00000 n /Descent -194 /CapHeight 683 gpAM^/)\,7,K[l6^+hoV`QWgGYJ2$kGYu4/j?L.$ngq16H+30^ZNVGb!RmqnXT[etcVV)PSmLHl ?rk.02GRK84em#Wb\ddR4LiU&bEm4P NpC(?>3/J"o3p8E/M3a!AWl6&FSa4$,CCUqQKp'p\#,cNs+dL!H]l! 0L.efqe^+Sr\BKjr9.AhoB"lc]1+eLfFKcGT;[""q^P#*I5%3fbY"!%A`_kU0T-qk]gik%l13#j /FirstChar 0 when is a Hilbert space) the concept of orthogonality can be used. 8 0 obj /FontName/IKFJSG+cmbx10 << /Type/Page !-hcUoHbOAa6GflYeC^$%=^ni;Gg#\pP!U@%G$B@L>+Z1j+*"c9P5~>endstream << 0000052747 00000 n E_eXsiMO>loIqH[L#OMB\[ /Differences[66/B 77/M 83/S/T 97/a 99/c 101/e 104/h/i 109/m 111/o 114/r/s/t 120/x] [K@/mh /FontName/OHKTCO+cmsy10 endobj Linear Algebra - Definition of Orthogonal Matrix What is Orthogonal Matrix? &)alB1g;2TV\,0a](Q\HG^J!^*6n /Parent 3 0 R 0000045427 00000 n endstream 4fUE74[=5q%hI] (;]O1*[JT6r0U4>ia^`!0^q[gC]g=-,DhYNUBj[YH\O1n@8$Hi(^!G GWEsk[N)Oia#-Ih<>?nbMWUWkciOUbTF2ToqI2_kLf?jaakIF:;8BoqImULjkaPED0_BrlB0M\R*DA1Aes@Sr$ZFS5=MT>L8, /Type/Font stream [jMh2`O)!ih-Xl;A@\>9S?23PGC51$B"3 /Name/F1 [b>sEIgV%MuaQX40`QL>H'EO!h+Spi>;@%qW3Rn[mpV])8-[YHsrPg /BaseEncoding/MacRomanEncoding 0000042180 00000 n 5X^gJZa96ec[6SC]c+T`Ri74Te_SmP%"G9$=\\/60+p=6et.gll>jJs'/*$/V3Qe&dn423 To find this orthogonal matrix , one uses the singular value decomposition (for which the entries of are non negative) = pCW9HbjV8m!otnJTmSC! /FontBBox[0 0 0 0] /Differences[45/hyphen/period 78/N 83/S 97/a/b/c/d/e/f/g/h/i 108/l/m/n/o/p 114/r/s/t 118/v 121/y/z] Suppose CTCb = 0 for some b. bTCTCb = (Cb)TCb = (Cb) •(Cb) = Cb 2 = 0. /Encoding 9 0 R 47 0 obj endobj /Differences[12/fi 40/parenleft/parenright 43/plus/comma/hyphen/period 48/zero/one/two/three/four/five/six/seven/eight/nine 59/semicolon 61/equal 65/A/B/C/D/E/F 72/H/I 76/L 78/N 80/P 83/S/T 87/W 91/bracketleft 93/bracketright 97/a/b/c/d/e/f/g/h/i/j 108/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z 124/emdash] endobj << /Length1 363 ?U,"*?Ii3AJ$Ch2G^H4l;6+h@DXh8OHZM22Tds4c4e /Type/Font Define , and then define to be the part of which is orthogonal to : Continue. Orthogonal Matrix Example. endobj /BaseFont/ILPSDF+cmsl10 where Q T {\displaystyle Q^{\mathrm {T} }} is the transpose of Q and I {\displaystyle I} is the identity matrix. /StemV 79 :7#1meuGiUX[@#BSZ6lSgjJL5!nr"uh*1f/?%*cbS?0Eug2!hb?16V^&;.%iJNto:Q1OkF/o.>/ .rqg-(V+S1(rEdj$=NU#rR;B,<49C\hA8JH"-1e&H%=,"FgH#.%pWb>]bSTg4PT;Xps_Kn">AQI /XHeight 0 ! 0000007721 00000 n hc#eL^q-Z)*^'lYQ0Y=t6>;"27pABT&Ni3dg?`M. 2gOCCs4mu@^9$`*m08NLrf_@+!=&W;DJ)(im1 Let T be a linear transformation from R^2 to R^2 given by the rotation matrix. /ItalicAngle 0 This completes the proof of Claim (1). Thus CTC is invertible. @q5'dQ_];1=n&:%HBZLI)LFm;r1<4b?Ja9[h-MK8AHG%;-]n3Uk.FkRZOh"W:RC]3'?L=A Example: Prove Q = \(\begin{bmatrix} cosZ & sinZ \\ -sinZ & cosZ\\ \end{bmatrix}\) is orthogonal matrix. 0000039533 00000 n endobj << /BaseEncoding/MacRomanEncoding << iR-K0::/a!Pj^l,[n=-'/TH5uF0Jkp'J0Li1A;. endobj jb4B*6U-l(WYnjZ"i$Q4WC[@'/-Pp;)8? /Length 2843 *+*E+.^HPe5QOk`W*5e^kR1l)Wt-A3V[4.`_pMf@FYQr=8#63i464bq?r)i?Gi`"tuH endobj /StemV 81 0000039329 00000 n kKN`uZ8-!7YpcA1cXJf@oj6A-]*bgD&:ru;7GmsYL! Glkg0leh LAG7d%_S5~>endstream << >> aY;YVA?K\#mM[i@'A7EX!.\Tu=EhBW[grU0RC0Yh'kK"n+_6];:)8TW /ItalicAngle -14 Vocabulary words: orthogonal set, orthonormal set. endobj We will now prove this with the following theorem. << /Subtype/Type1 >> /Differences[49/one/two] >> Theorem Let A be an m × n matrix, let W = Col ( A ) , and let x be a vector in R m . /Flags 262148 0000035585 00000 n !s=#Yi79rR? =G$m /Type/Font /Descent -117 [1SmT*tj#IV9@!13d:^q*VGC[\"-+pr1[%L3KF==*3l@B/=CK=Xn0Yuq5(']i /FontName/KNJGSY+cmmi7 /Subtype/Type1C I43DT]T[uAo'\@&F0$,pLEPUEW&;I!K>GRlf\nBAC7!/`RPgt6'3E6UAXR$TET.>p8IA#E/%d'O /Type/FontDescriptor Z2uSP8=4XbrZB?Xl4ap*kLg6UI;1"b(H&X/W$Tm71h?dc]8*>3[oH6Ue5U>^TbM/4W)'Z4lM3-`KbscD:/D>Y7s%5t`/,ofDJI /Subtype/Type1 UKbGDk=s8Dkha2t/M4rRfGn_QjU&m?ik>(S)8U'>?TRMe-b?Gb.c5^dZ+t85#cp'_nma&^VObo?N0Y+/+1!g"AkZSQG[dl"A33GZY#88aGMc12u3@HU6&F,pWjgj+> /StemV 72 /Resources<> @ stream << 0oE9:Jg+'(HbR&&Xkc)Rf#j'?l;Lb^Z\[/F1%;u8\f`r>;55M /Name/F3 Therefore N(A) = S⊥, where S is the set of rows of A. 28 0 obj ';1["GrY/-7'^4(\XOS6 /Type/Encoding endobj >> Psh*>:D-s$PfHi8jit,DZmSbVn7b3&n3U^.c5H[OZUZo0W>b_Qm&ra4,'? endobj /MediaBox[0 0 612 792] Write     cos sin,)) where) is the angle between the positive … hUU02J)%dk_9(OhCAF+`K+b:J7L>=tQuZRhWh?Xh4?%7S >> 0000043898 00000 n 3DDGV8s;<8G4:*&\KMfTr&$o)/bsa9+1DuVDJH=q4sY"\Z"2r'_@SZco41#f^q46qc$jaNriF1U /Type/Encoding /Name/F12 /Name/F11 ]JD:qn1%8e=lJYC>]-AIk?`f`;PnM`c2AP^)lZG-KDfmp'W@;&.AU8*OAd8OAT[Zor#La&aYu4i/O-il2e($al>ri*T8jG6A-GTOQT\W9rK?^$&I="+;L*mA-RA^OXXFEG0p5E!c'W#?`m^#!_6bKfaL7P&39Z*CD"?<2mQH,=(/Es*\a 6 0 obj It is a beautiful story which carries the beautiful name the spectral theorem: Theorem 1 (The spectral theorem). /BaseEncoding/MacRomanEncoding << << 14 0 obj /Encoding 17 0 R gB74_kbK*CgYVqCqXDj243[HBA>cCb)Y\!=ERc,T[E$BBpP"%RG]sX8.>iendstream We will build our orthogonal basis by orthogonalizing one vector at a time. /BaseEncoding/MacRomanEncoding << A matrix P is called orthogonal if its columns form an orthonormal set and call a matrix A orthogonally diagonalizable if it can be diagonalized by D = P-1 AP with P an orthogonal matrix. See pages that link to and include this page. /BaseEncoding/MacRomanEncoding << /Length 606 .\s5'AUg:r!meg! /LastChar 122 /Subtype/Type1C HU7Mh,7Z/\&MSN20$=tce80%MYIK7Cf-eo60H7-qbD52Dmc1daW%sJshFsT5%CIJ5V5N.W\(*bJ /FontBBox[0 0 0 0] obpPO)l(qA2/BTEqa]1K!#L4F8an&D9E-T4f3)Dng8G-P?'Gqk!54qUIg"+9BEN! /Filter [/ASCII85Decode/FlateDecode] /Subtype/Type1C >> /Descent -194 stream << >> /CapHeight 655 45 0 obj /LastChar 121 H2>aPf&jk+$Y*W9X;9G'>T]uVRI%&fJ9c>a0/J_+jFVkJng@+'&pXVT.! Answer Save. /FontBBox[0 0 0 0] /Widths[530 530] 0000022465 00000 n endobj Let $\lambda$ be an eigenvalue of $A$ and let $\mathbf{v}$ be a corresponding eigenvector. _diCk*'3h4id.Fi/MXL!8b>B/:YNgJ^ft,i"L,J+Y,38&-!-_&QX(kr*BGl/4a+$6NY.#i)(t>2 &0jgFj<7m>cA^%EOsMkZ]SGK1q+Z%5Lud<4]`Jk$h2@\#2SZP$atQ>rc1cRFd^h7UHT%WoZ>"H)-oDgUQ$bW:5;g%u startxref << C_O+!4`e!QZG28]@)eK7#=93=4V&B@Vat$u@k?k\qGs*T$XS=uG.ITts[/TXS`L7Dk'+6D>S]'&>YG=fQ+=dbkF'oQugE%5R] << Theorem 3.2. 8)br&&Of]LEr=8bT^1CG]q! 4 0 obj K$m8YjG_`e#\d8;]XGPm:u/FPd:LE)/oNM4SG`Zrh:e^+UYEAAWf;FL$K+"-_Sf/-$L2NEDSCH% K@#mo\`\GDcf94SdVb+W;_A[60Q*hU!VTsIZbqlZO;q2;(F8+Yi$*$1q1q92C>] /Type/FontDescriptor ?4AfX,;/.Y+#h'"hBSg$2V242f#:b!E`l$7?3IC%Gb-9r!`JX06RX)(\s\1$,VO^H.m/9eh29sb@FCJF 0000046172 00000 n 6Y_`iL"EI.GqQDYg]+,B.P8lic"Ft>=FN+$Slnn+,mq3Wi>.\[R*6#]SX,>N`:.N$TZBnS6k2t-Jm_aUWtj38)LWWEQ2Yf6Xd9%9bBI+)XC /Name/F7 %���� .+E)0]iAu[a0JM]Zqt?G8*p5m9UGu:S-DJVjY!o%>;51SF2>t\=V34I&U8H5MdSY$f#%^0+^Tk0 ]X-'UO?jDCUOB;DUZ(e@Q+D4 The following theorem gives us a relatively nice formula to use. << =G=FiCNpP)R]WGY&OOno>+"U+.NCoJ@-"%1U?Dk2(iBC0!KfHX84_q#\TL6%^/rihI8B$TlZD=@ << stream >> ^[DElHP(+d^?a?g8EOf&c7Y)drPLMta7JT"J=uFa5;Iqrd],fumXMQ=cDJG_TpP-6&X\YsPeF*k /Ascent 683 @Nt'L^ij?8pZ,II*H]O3A/\->Is%W"q\\6?OgF[8j'bsNi@)&Mq65eSk,?j#%g>o !ft0hZ.Op)>+iQhmfgin %Mb^2oL>jX%""q[>b8la^7snqj,%? mSh,(IUcQ=:r9g5Ul)662:'4">WeRsc.+QeG)0^(9_I$AGeeH\3aB(_`5LB'Mo0n/WV92Ca(c-\ !hQ+%j2m5]BGr`+)Kn>[Lq$-MmO1fRkDD8%)^>$MqM8s.b)VFnR@8oQIUmMD\ig:L7^nS;C#=RY 19 0 obj ;gH/dYjb`=4f%/2-f32lluse6,VdSH'g)E77stdT]:Hf!1pW;@>ntT 0000021828 00000 n ? >> /Type/Font 24QOJYT;OhTnsQfBb`0N&>b.+f`s^(TgphJ)=h9tU'')&7UsgeDX"0_F0?Yl6LNKbIIM=uNMAc\ H4N;!b=_[WD:V[YjpRlgWbe^NISqe6G5+L=]/G',\S`MZA43koj27d;>CTi.bIX5PR3hGNNHAJ\ 55 0 obj << The vector $\vec{w_1}$ has a special name, which we will formally define as follows. /Differences[105/i/j 110/n] A(8o;U^F^&Z6rdlG:S.&!B%N7W#;G'mq*#;D6O;,,K!VenVj\\'b>^.r]2b0g( /XHeight 705 >> endobj endobj >> /Type/Encoding R_o!>Jq=UufktK5>>O4'-"lbbU*9nMX^c1L/Hp7W@/_db((1j*[V!Ri#-P=H$f$nj-ErShOS&Y"N! endobj H7B%XS>TbT#_iBMmA$:$IZNAonaiE'U&D1#XF`27WWq\INX06k2n-@B9]2>?-3%')fsA_2_4bN" !9MrT-9ot/Q@"S4X=CrlrsYM%qpj*@dfqsU!GC48jPGSM$5u=O(! /CapHeight 683 /BaseEncoding/MacRomanEncoding /Subtype/Type1C << << Xlqb+X9WC?qY`9'$g&][DTH9>gRn"ScqI0$RoRZ>@bl/hf#U4`b[-aumFe$+7("UF! >> ),91^g!hlT:+)3M)WDl>%-5eAcPQ2e!BP),C47jE;t!Xao_F)f Yt#eK+/f>YJX1eh(/"@%>,JKXSXU?3lI9=T`Sj-Ed00cbW+[\s9Q$QJIZb^pUcjnK 0000050946 00000 n $d>8X_4:b_8.'31H6P;.bu>4tYY*ToQ? /FontDescriptor 59 0 R

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