869.4 818.1 830.6 881.9 755.6 723.6 904.2 900 436.1 594.4 901.4 691.7 1091.7 900 /BaseFont/UJZCKN+CMR8 /Name/F4 Cb = 0 b = 0 since C has L.I. 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 /Length 2119 /Type/Font /BaseFont/AWSEZR+CMTI10 A linear transformation from Rn to Rn is called orthogonal if it preserves lengths. They might just kind of rotate them around or shift them a little bit, but it doesn't change the angles between them. /Name/F2 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 694.5 295.1] /BaseFont/CYTIPA+CMEX10 endobj /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 465 322.5 384 636.5 500 277.8 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 Let us now rotate u1 and u2 3gis thus an orthogonal set of eigenvectors of A. Corollary 1. 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /BaseFont/OHWPLS+CMMI8 /Encoding 7 0 R /Type/Font 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 T8‚8 T TœTSince is square and , we have " X "œ ÐTT Ñœ ÐTTќРTÑÐ TќРTÑ Tœ„"Þdet det det det det , so det " X X # Theorem Suppose is orthogonal. 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 7. 2. 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 /FirstChar 33 /Widths[777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 32 0 obj As an application, we prove that every 3 by 3 orthogonal matrix has always 1 as an eigenvalue. Then to summarize, Theorem. /Type/Font 10 0 obj if det , then the mapping is a rotationñTœ" ÄTBB 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 /FontDescriptor 18 0 R As we will now prove, this set is a subgroup of O(n) and it is called the special orthogonal group, denoted SO(n). 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 Show that QQT = I. endobj 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 >> endobj Orthogonal matrices and orthonormal sets An n£n real-valued matrix A is said to be an orthogonal matrix if ATA = I; (1) or, equivalently, if AT = A¡1. 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 /Name/F6 Orthogonal Matrices Let Q be an n×n matrix. 826.4 295.1 531.3] 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 Products and inverses of orthogonal matrices a. 277.8 972.2 625 625 625 625 416.7 479.2 451.4 625 555.6 833.3 555.6 555.6 538.2 625 Note. /BaseFont/IHGFBX+CMBX10 Proof thesquareddistanceofb toanarbitrarypointAx inrange„A”is kAx bk2 = kA„x xˆ”+ Axˆ bk2 (wherexˆ = ATb) = kA„x xˆ”k2 + kAxˆ bk2 +2„x xˆ”TAT„Axˆ b” = kA„x xˆ”k2 + kAxˆ bk2 = kx xˆk2 + kAxˆ bk2 kAxˆ bk2 withequalityonlyifx = xˆ line3followsbecauseAT„Axˆ b”= xˆ ATb = 0 line4followsfromATA = I Orthogonalmatrices 5.18 /Differences[33/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi Thus, a matrix is orthogonal … /LastChar 196 /BaseFont/NSPEWR+CMSY8 << /Subtype/Type1 >> 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 endobj /Subtype/Type1 Proof. 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 /FirstChar 33 >> /Encoding 7 0 R 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1 . 2& where7 4 is the smallest non-zerosingular value. The product AB of two orthogonal n £ n matrices A and B is orthogonal. Orthogonal Matrices Now we move on to consider matrices analogous to the Qshowing up in the formula for the matrix of an orthogonal projection. /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 Orthogonal Matrices#‚# Suppose is an orthogonal matrix. /Type/Font 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 812.5 965.3 784.7 965.3 816 694.4 895.8 809 805.6 1152.8 805.6 805.6 763.9 352.4 Note that for a full rank square matrix, !3) is the same as !0!). Orthogonal Transformations and Matrices Linear transformations that preserve length are of particular interest. << endobj ��^׎+��������Em�\�+�G���2��cP���A�d�E�W�H�76)"�. 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 777.8 1000 1000 1000 1000 1000 1000 777.8 777.8 555.6 722.2 666.7 722.2 722.2 666.7 To determine if a matrix is orthogonal, we need to multiply the matrix by it's transpose, and see if we get the identity matrix., Since we get the identity matrix, then we know that is an orthogonal matrix. 40 0 obj 0 0 0 0 0 0 691.7 958.3 894.4 805.6 766.7 900 830.6 894.4 830.6 894.4 0 0 830.6 670.8 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 /Subtype/Type1 Eigenvalues and Eigenvectors The eigenvalues and eigenvectors of a matrix play an important part in multivariate analysis. /Encoding 7 0 R 21 0 obj 7 0 obj We know that any subspace of Rn has a basis. << The di erence now is that while Qfrom before was not necessarily a square matrix, here we consider ones which are square. So orthogonal vectors make things much easier. /FirstChar 33 endobj 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 << /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 /FirstChar 33 3. Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. /Subtype/Type1 16 0 obj �4���w��k�T�zZ;�7��‹ �����އt2G��K���QiH��ξ�x�H��u�iu�ZN�X;]O���DŽ�MD�Z�������y!�A�b�������؝� ����w���^�d�1��&�l˺��I`/�iw��������6Yu(j��yʌ�a��2f�w���i�`�ȫ)7y�6��Qv�� T��e�g~cl��cxK��eQLl�&u�P�=Z4���/��>� /Type/Font /FontDescriptor 18 0 R In the same way, the inverse of the orthogonal matrix… Suppose CTCb = 0 for some b. bTCTCb = (Cb)TCb = (Cb) •(Cb) = Cb 2 = 0. /Subtype/Type1 /Subtype/Type1 /Name/F3 The vectors u1 =(1,0) and u2 =(0,1) form an orthonormal basis B = {u1,u2}. 26 0 obj /Subtype/Type1 527.1 496.5 680.6 604.2 909.7 604.2 604.2 590.3 687.5 1375 687.5 687.5 687.5 0 0 /Widths[354.2 625 1041.7 625 1041.7 937.5 347.2 486.1 486.1 625 972.2 347.2 416.7 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 0 0 0 0 722.2 555.6 777.8 666.7 444.4 666.7 777.8 777.8 777.8 777.8 222.2 388.9 777.8 /LastChar 196 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 0 642.9 885.4 806.2 736.8 >> endobj << William Ford, in Numerical Linear Algebra with Applications, 2015. 255/dieresis] 511.1 575 1150 575 575 575 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 0 694.5 295.1] /Filter[/FlateDecode] /Type/Font 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 /LastChar 196 ORTHOGONAL MATRICES 10.1. 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 777.8 777.8 777.8 888.9 888.9 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 We prove that eigenvalues of orthogonal matrices have length 1. /LastChar 196 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 /Length 625 >> /BaseFont/WOVOQW+CMMI10 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 /BaseFont/QQXJAX+CMMI8 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 endobj 500 500 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 0 625 833.3 /Type/Encoding /FirstChar 33 Definition An matrix is called 8‚8 E orthogonally diagonalizable if there is an orthogonal matrix and a diagonal matrix for which Y H EœYHY ÐœYHY ÑÞ" X Thus, an orthogonally diagonalizable matrix is a special kind of diagonalizable matrix… Projection '' matrix P such that P~x= ~v ( if ~x,,! 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