> 13 0 obj 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 Thesis, 2004 Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. 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 19 0 obj endobj A matrix P is orthogonal if P T P = I, or the inverse of P is its transpose. 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 We know that any subspace of Rn has a basis. The transpose of the orthogonal matrix is also orthogonal. 869.4 818.1 830.6 881.9 755.6 723.6 904.2 900 436.1 594.4 901.4 691.7 1091.7 900 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 /Encoding 7 0 R >> >> << /Filter[/FlateDecode] Alternatively, a matrix is orthogonal if and only if its columns are orthonormal, meaning they are orthogonal and of unit length. 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 /Subtype/Type1 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 Orthogonal matrix is an important matrix in linear algebra, it is also widely used in machine learning. /BaseFont/UPABUT+CMSY8 /Length 2119 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 7 0 obj 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 Properties of real symmetric matrices I Recall that a matrix A 2Rn n is symmetric if AT = A. I For real symmetric matrices we have the following two crucial properties: I All eigenvalues of a real symmetric matrix are real. 458.3 381.9 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 381.9 They might just kind of rotate them around or shift them a little bit, but it doesn't change the angles between them. So, given a matrix M, ﬁnd the matrix Rthat minimizes M−R 2 F, subject to RT R = I, where the norm chosen is the Frobenius norm, i.e. >> %PDF-1.2 A matrix V that satisﬁes equation (3) is said to be orthogonal. >> endobj /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 625 1062.5 1201.4 972.2 277.8 625] 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 /FontDescriptor 15 0 R /Subtype/Type1 /LastChar 196 5) Norm of the pseudo-inverse matrix The norm of the pseudo-inverse of a (×*matrix is:!3=.-3,#!3)=! /Type/Font >> endobj /Subtype/Type1 /Filter[/FlateDecode] /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 833.3 805.6 819.4 798.6 888.9 777.8 743.1 833.3 812.5 319.4 576.4 840.3 708.3 1020.8 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 << /Encoding 7 0 R The product AB of two orthogonal n £ n matrices A and B is orthogonal. /Name/F1 The orthonormal set can be obtained by scaling all vectors in the orthogonal set of Lemma 5 to have length 1. >> If Q is square, then QTQ = I tells us that QT = Q−1. A great example is projecting onto a subspace. 23 0 obj /BaseFont/CYTIPA+CMEX10 Orthogonal Projection Matrix •Let C be an n x k matrix whose columns form a basis for a subspace W = −1 n x n Proof: We want to prove that CTC has independent columns. /Type/Font /FirstChar 33 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 Note that for a full rank square matrix, !3) is the same as !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 0 663.6 885.4 826.4 736.8 /Widths[392.4 687.5 1145.8 687.5 1183.3 1027.8 381.9 534.7 534.7 687.5 1069.5 381.9 Thus, if matrix A is orthogonal, then is A T is also an orthogonal matrix. /Type/Font i.e. That is, T is orthogonal if jjT(x)jj= jjxjjfor all x in Rn. 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 endobj >> 255/dieresis] Proof. 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 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 �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���/��>� /Encoding 20 0 R >> /FontDescriptor 18 0 R /Type/Encoding Introduction Definition. 361.1 635.4 927.1 777.8 1128.5 899.3 1059 864.6 1059 897.6 763.9 982.6 894.1 888.9 /FirstChar 33 /Name/F9 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 /Subtype/Type1 /BaseFont/CXMPOE+CMSY10 De nitions and Theorems from 5.3 Orthogonal Transformations and Matrices, the Transpose of a Matrix De nition 1. The following are equivalent characterizations of an orthogonal matrix Q: If Ais the matrix of an orthogonal transformation T, then AAT is the identity matrix. T8‚8 T TœTSince is square and , we have " X "œ ÐTT Ñœ ÐTTÑœÐ TÑÐ TÑœÐ TÑ Tœ„"Þdet det det det det , so det " X X # Theorem Suppose is orthogonal. 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 The collection of the orthogonal matrix of order n x n, in a group, is called an orthogonal group and is denoted by ‘O’. b.The inverse A¡1 of an orthogonal n£n matrix A is orthogonal. \$3(JH/���%�%^h�v�9����ԥM:��6�~���'�ɾ8�>ݕE��D�G�&?��3����]n�}^m�]�U�e~�7��qx?4�d.њ��N�`���\$#�������|�����߁��q �P����b̠D�>�� 16 0 obj %PDF-1.2 694.5 295.1] << /Type/Font /FontDescriptor 15 0 R /LastChar 196 275 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 /Name/F3 1062.5 826.4] 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 1062.5 826.4] /Type/Font 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 Let us now rotate u1 and u2 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 This is valid for any matrix, regardless of the shape or rank. /Type/Font /Subtype/Type1 )��R\$���_W?՛����i�ڷ}xl����ڮ�оo��֏諭k6��v���. 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 /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 3gis thus an orthogonal set of eigenvectors of A. Corollary 1. 1322.9 1069.5 298.6 687.5] 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 stream Let A be an n nsymmetric matrix. << /Subtype/Type1 Consider the euclidean space R2 with the euclidean inner product. 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 /FirstChar 0 /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 Every n nsymmetric matrix has an orthonormal set of neigenvectors. 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 500 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 500 500 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 endobj >> /FirstChar 33 /Type/Font Proof In part (a), the linear transformation T(~x) = AB~x preserves length, because kT(~x)k = kA(B~x)k = kB~xk = k~xk. The most desirable class of matrices … << 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 >> /BaseFont/OHWPLS+CMMI8 10 0 obj endobj << << The vectors u1 =(1,0) and u2 =(0,1) form an orthonormal basis B = {u1,u2}. 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 Orthogonal matrix • 2D example: rotation matrix nothing. 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 /Name/F2 525 768.9 627.2 896.7 743.3 766.7 678.3 766.7 729.4 562.2 715.6 743.3 743.3 998.9 /FirstChar 33 /LastChar 196 /Type/Font De nition A matrix Pis orthogonal if P 1 = PT. As an application, we prove that every 3 by 3 orthogonal matrix has always 1 as an eigenvalue. 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 575 1041.7 1169.4 894.4 319.4 575] /Name/F4 /FirstChar 33 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 Figure 4 illustrates property (a). /Type/Font /FontDescriptor 25 0 R Eigenvalues and Eigenvectors The eigenvalues and eigenvectors of a matrix play an important part in multivariate analysis. Note. A linear transform T: R n!R is orthogonal if for all ~x2Rn jjT(~x)jj= jj~xjj: Likewise, a matrix U2R n is orthogonal if U= [T] for T an orthogonal trans-formation. /BaseFont/QQXJAX+CMMI8 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 /LastChar 196 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 endobj Is the product of k > 2 orthogonal matrices an orthogonal matrix? 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 Exercise 3.5 Let Q be an orthogonal matrix, i.e., QTQ = I. /FontDescriptor 31 0 R Products and inverses of orthogonal matrices a. >> /FirstChar 33 /LastChar 196 >> A square orthonormal matrix Q is called an orthogonal matrix. Thus, a matrix is orthogonal … endobj /FontDescriptor 12 0 R /FontDescriptor 34 0 R 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 Show that the product U1U2 of two orthogonal matrices is an orthogonal matrix. 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 /BaseFont/MITRMO+MSBM10 endobj /FontDescriptor 12 0 R 10 0 obj /Subtype/Type1 /LastChar 127 endobj 0 708.3 1041.7 972.2 736.1 833.3 812.5 902.8 972.2 902.8 972.2 0 0 902.8 729.2 659.7 It turns xڭUMo�@��Wp)���b���[ǩ�ƖnM�Ł 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 0 777.8 1145.8 1069.5 /Length 625 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 /Subtype/Type1 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] On Orthogonal Matrices Majid Behbahani Department of Mathematics and Computer Science University of Lethbridge M. Sc. 29 0 obj Fact. endobj 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] /BaseFont/UJZCKN+CMR8 << ORTHOGONAL MATRICES 10.1. ��^׎+��������Em�\�+�G���2��cP���A�d�E�W�H�76)"�. 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 /BaseFont/IHGFBX+CMBX10 endobj 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 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. /LastChar 196 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 << 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 /Type/Font Let C be a matrix with linearly independent columns. So let ~v 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1 . Explanation: . /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 /Encoding 7 0 R /Name/F1 Matrices of eigenvectors (discussed below) are orthogonal matrices. 6. 298.6 336.8 687.5 687.5 687.5 687.5 687.5 888.9 611.1 645.8 993.1 1069.5 687.5 1170.1 /Subtype/Type1 William Ford, in Numerical Linear Algebra with Applications, 2015. 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 Now we prove an important lemma about symmetric matrices. /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 IfTœ +, -. 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 matrices”. 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 >> 1250 625 625 625 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 /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 If a matrix A is an orthogonal matrix, it shoud be n*n. The feature of an orthogonal matrix A. /BaseFont/BBRNJB+CMR10 7. /BaseFont/EXOVXJ+LCMSS8 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 35 0 obj 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 /Type/Encoding Learning Goals: learn about orthogonal matrices and their use in simplifying the least squares problem, and the QR factorization and its speed improvements to the least squares problem. 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 In fact, we can nd a nice formula for P. Setup: Our strategy will be to create P rst and then use it to verify all the above statements. 777.8 777.8 777.8 777.8 777.8 1000 1000 777.8 666.7 555.6 540.3 540.3 429.2] /Name/F4 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 /Name/F6 << /Type/Font /BaseFont/WOVOQW+CMMI10 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 /LastChar 196 Theorem 1.9. 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 626.7 420.1 680.6 680.6 298.6 336.8 642.4 298.6 1062.5 680.6 687.5 680.6 680.6 454.9 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 0 458.3 458.3 416.7 416.7 /Name/F5 /FontDescriptor 28 0 R 40 0 obj 347.2 625 625 625 625 625 625 625 625 625 625 625 347.2 347.2 354.2 972.2 590.3 590.3 2& where7 4 is the smallest non-zerosingular value. /Subtype/Type1 >> 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 If A 1 = AT, then Ais the matrix of an orthogonal transformation of Rn. Orthogonal Matrices Now we move on to consider matrices analogous to the Qshowing up in the formula for the matrix of an orthogonal projection. Lemma 6. << 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 8. /FontDescriptor 22 0 R Thus CTC is invertible. /FirstChar 33 277.8 500] 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 The set of elements in O(n) with determinant +1 is the set of all proper rotations on Rn. << 659.7 1006.9 1006.9 277.8 312.5 625 625 625 625 625 805.6 555.6 590.3 902.8 972.2 /Differences[0/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/exclam/quotedblright/numbersign/sterling/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 694.5 295.1] << /BaseFont/AWSEZR+CMTI10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 627.2 817.8 766.7 692.2 664.4 743.3 715.6 A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. 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 In the same way, the inverse of the orthogonal matrix… I Eigenvectors corresponding to distinct eigenvalues are orthogonal. /Type/Font 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 Cb = 0 b = 0 since C has L.I. endobj 7 0 obj << This matrix is called the identity,denotedI. 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 The product of two orthogonal matrices (of the same size) is orthogonal. 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 The transpose of an orthogonal matrix is orthogonal. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 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… 1270.8 888.9 888.9 840.3 416.7 687.5 416.7 687.5 381.9 381.9 645.8 680.6 611.1 680.6 Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 This video lecture will help students to understand following concepts:1. /BaseFont/NSPEWR+CMSY8 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 Orthogonal matrices are the most beautiful of all matrices. View Orthogonal_Matrices.pdf from MATH 2418 at University of Texas, Dallas. /Encoding 7 0 R A square matrix A with real entries and satisfying the condition A−1 = At is called an orthogonal matrix. 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 >> /LastChar 196 Orthogonal matrices are very important in factor analysis. 255/dieresis] /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 ~X, ~v, and w~are as above ) that any subspace of Rn of! 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# orthogonal matrix pdf

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>> /LastChar 196 endobj 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 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 0 0 0 0 0 642.9 885.4 806.2 736.8 Example 10.1.1. /FirstChar 33 /Subtype/Type1 /LastChar 196 That is, for all ~x, jjU~xjj= jj~xjj: EXAMPLE: R Orthogonal matrix with properties and examples.2. if det , then the mapping is a rotationñTœ" ÄTBB Overview. << Then to summarize, Theorem. /FontDescriptor 9 0 R If T(x) = Ax is an orthogonal linear transformation, we say A is an orthogonal matrix. Suppose CTCb = 0 for some b. bTCTCb = (Cb)TCb = (Cb) •(Cb) = Cb 2 = 0. A linear transformation from Rn to Rn is called orthogonal if it preserves lengths. 26 0 obj endobj 666.7 722.2 722.2 1000 722.2 722.2 666.7 1888.9 2333.3 1888.9 2333.3 0 555.6 638.9 9. Example using orthogonal change-of-basis matrix to find transformation matrix. 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 /Widths[350 602.8 958.3 575 958.3 894.4 319.4 447.2 447.2 575 894.4 319.4 383.3 319.4 If an element of the diagonal is zero, then the associated axis is annihilated. 10 ORTHOGONALITY 7 Therefore, c = 5 7 and d = 6 7 and the best ﬁtting line is y = 5 7 + 6 7x, which is the line shown in the graph. There is an \orthogonal projection" matrix P such that P~x= ~v(if ~x, ~v, and w~are as above). 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. 21 0 obj 13 0 obj 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] 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 32 0 obj An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.Although we consider only real matrices here, the definition can be used for matrices with entries from any field.However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. 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 0 826.4 295.1 826.4 531.3 826.4 orthogonal matrix is a square matrix with orthonormal columns. 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 Both Qand T 0 1 0 1 0 0 are orthogonal matrices, and their product is the identity. << 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 We prove that eigenvalues of orthogonal matrices have length 1. 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 /Type/Font 777.8 777.8 777.8 500 277.8 222.2 388.9 611.1 722.2 611.1 722.2 777.8 777.8 777.8 x��Z[�ܶ~���`1�_��E��m������7ί�!)J���ٛ�eG�y.�΅R��B! 3. (We could tell in advance that the matrix equation Ax = b has no solution since the points are not collinear. /Subtype/Type1 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 /Name/F2 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 /FontDescriptor 18 0 R Recall that Q is an orthogonal matrix if it satisfies QT = Q−1 . 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 0 826.4 295.1 826.4 531.3 826.4 /Name/F3 /Subtype/Type1 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 0 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 FUNDAMENTALS OF LINEAR ALGEBRA James B. Carrell carrell@math.ubc.ca (July, 2005) 2 1 ORTHOGONAL MATRICES In matrix form, q = VTp : (2) Also, we can collect the n2 equations vT i v j = ˆ 1 if i= j 0 otherwise into the following matrix equation: VTV = I (3) where Iis the n nidentity matrix. Hence all orthogonal matrices must have a determinant of ±1. >> 13 0 obj 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 Thesis, 2004 Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. 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 19 0 obj endobj A matrix P is orthogonal if P T P = I, or the inverse of P is its transpose. 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 We know that any subspace of Rn has a basis. The transpose of the orthogonal matrix is also orthogonal. 869.4 818.1 830.6 881.9 755.6 723.6 904.2 900 436.1 594.4 901.4 691.7 1091.7 900 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 /Encoding 7 0 R >> >> << /Filter[/FlateDecode] Alternatively, a matrix is orthogonal if and only if its columns are orthonormal, meaning they are orthogonal and of unit length. 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 /Subtype/Type1 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 Orthogonal matrix is an important matrix in linear algebra, it is also widely used in machine learning. /BaseFont/UPABUT+CMSY8 /Length 2119 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 7 0 obj 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 Properties of real symmetric matrices I Recall that a matrix A 2Rn n is symmetric if AT = A. I For real symmetric matrices we have the following two crucial properties: I All eigenvalues of a real symmetric matrix are real. 458.3 381.9 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 381.9 They might just kind of rotate them around or shift them a little bit, but it doesn't change the angles between them. So, given a matrix M, ﬁnd the matrix Rthat minimizes M−R 2 F, subject to RT R = I, where the norm chosen is the Frobenius norm, i.e. >> %PDF-1.2 A matrix V that satisﬁes equation (3) is said to be orthogonal. >> endobj /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 625 1062.5 1201.4 972.2 277.8 625] 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 /FontDescriptor 15 0 R /Subtype/Type1 /LastChar 196 5) Norm of the pseudo-inverse matrix The norm of the pseudo-inverse of a (×*matrix is:!3=.-3,#!3)=! /Type/Font >> endobj /Subtype/Type1 /Filter[/FlateDecode] /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 833.3 805.6 819.4 798.6 888.9 777.8 743.1 833.3 812.5 319.4 576.4 840.3 708.3 1020.8 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 << /Encoding 7 0 R The product AB of two orthogonal n £ n matrices A and B is orthogonal. /Name/F1 The orthonormal set can be obtained by scaling all vectors in the orthogonal set of Lemma 5 to have length 1. >> If Q is square, then QTQ = I tells us that QT = Q−1. A great example is projecting onto a subspace. 23 0 obj /BaseFont/CYTIPA+CMEX10 Orthogonal Projection Matrix •Let C be an n x k matrix whose columns form a basis for a subspace W = −1 n x n Proof: We want to prove that CTC has independent columns. /Type/Font /FirstChar 33 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 Note that for a full rank square matrix, !3) is the same as !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 0 663.6 885.4 826.4 736.8 /Widths[392.4 687.5 1145.8 687.5 1183.3 1027.8 381.9 534.7 534.7 687.5 1069.5 381.9 Thus, if matrix A is orthogonal, then is A T is also an orthogonal matrix. /Type/Font i.e. That is, T is orthogonal if jjT(x)jj= jjxjjfor all x in Rn. 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 endobj >> 255/dieresis] Proof. 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 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 �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���/��>� /Encoding 20 0 R >> /FontDescriptor 18 0 R /Type/Encoding Introduction Definition. 361.1 635.4 927.1 777.8 1128.5 899.3 1059 864.6 1059 897.6 763.9 982.6 894.1 888.9 /FirstChar 33 /Name/F9 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 /Subtype/Type1 /BaseFont/CXMPOE+CMSY10 De nitions and Theorems from 5.3 Orthogonal Transformations and Matrices, the Transpose of a Matrix De nition 1. The following are equivalent characterizations of an orthogonal matrix Q: If Ais the matrix of an orthogonal transformation T, then AAT is the identity matrix. T8‚8 T TœTSince is square and , we have " X "œ ÐTT Ñœ ÐTTÑœÐ TÑÐ TÑœÐ TÑ Tœ„"Þdet det det det det , so det " X X # Theorem Suppose is orthogonal. 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 The collection of the orthogonal matrix of order n x n, in a group, is called an orthogonal group and is denoted by ‘O’. b.The inverse A¡1 of an orthogonal n£n matrix A is orthogonal. \$3(JH/���%�%^h�v�9����ԥM:��6�~���'�ɾ8�>ݕE��D�G�&?��3����]n�}^m�]�U�e~�7��qx?4�d.њ��N�`���\$#�������|�����߁��q �P����b̠D�>�� 16 0 obj %PDF-1.2 694.5 295.1] << /Type/Font /FontDescriptor 15 0 R /LastChar 196 275 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 /Name/F3 1062.5 826.4] 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 1062.5 826.4] /Type/Font 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 Let us now rotate u1 and u2 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 This is valid for any matrix, regardless of the shape or rank. /Type/Font /Subtype/Type1 )��R\$���_W?՛����i�ڷ}xl����ڮ�оo��֏諭k6��v���. 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 /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 3gis thus an orthogonal set of eigenvectors of A. Corollary 1. 1322.9 1069.5 298.6 687.5] 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 stream Let A be an n nsymmetric matrix. << /Subtype/Type1 Consider the euclidean space R2 with the euclidean inner product. 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 /FirstChar 0 /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 Every n nsymmetric matrix has an orthonormal set of neigenvectors. 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 500 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 500 500 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 endobj >> /FirstChar 33 /Type/Font Proof In part (a), the linear transformation T(~x) = AB~x preserves length, because kT(~x)k = kA(B~x)k = kB~xk = k~xk. The most desirable class of matrices … << 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 >> /BaseFont/OHWPLS+CMMI8 10 0 obj endobj << << The vectors u1 =(1,0) and u2 =(0,1) form an orthonormal basis B = {u1,u2}. 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 Orthogonal matrix • 2D example: rotation matrix nothing. 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 /Name/F2 525 768.9 627.2 896.7 743.3 766.7 678.3 766.7 729.4 562.2 715.6 743.3 743.3 998.9 /FirstChar 33 /LastChar 196 /Type/Font De nition A matrix Pis orthogonal if P 1 = PT. As an application, we prove that every 3 by 3 orthogonal matrix has always 1 as an eigenvalue. 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 575 1041.7 1169.4 894.4 319.4 575] /Name/F4 /FirstChar 33 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 Figure 4 illustrates property (a). /Type/Font /FontDescriptor 25 0 R Eigenvalues and Eigenvectors The eigenvalues and eigenvectors of a matrix play an important part in multivariate analysis. Note. A linear transform T: R n!R is orthogonal if for all ~x2Rn jjT(~x)jj= jj~xjj: Likewise, a matrix U2R n is orthogonal if U= [T] for T an orthogonal trans-formation. /BaseFont/QQXJAX+CMMI8 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 /LastChar 196 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 endobj Is the product of k > 2 orthogonal matrices an orthogonal matrix? 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 Exercise 3.5 Let Q be an orthogonal matrix, i.e., QTQ = I. /FontDescriptor 31 0 R Products and inverses of orthogonal matrices a. >> /FirstChar 33 /LastChar 196 >> A square orthonormal matrix Q is called an orthogonal matrix. Thus, a matrix is orthogonal … endobj /FontDescriptor 12 0 R /FontDescriptor 34 0 R 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 Show that the product U1U2 of two orthogonal matrices is an orthogonal matrix. 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 /BaseFont/MITRMO+MSBM10 endobj /FontDescriptor 12 0 R 10 0 obj /Subtype/Type1 /LastChar 127 endobj 0 708.3 1041.7 972.2 736.1 833.3 812.5 902.8 972.2 902.8 972.2 0 0 902.8 729.2 659.7 It turns xڭUMo�@��Wp)���b���[ǩ�ƖnM�Ł 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 0 777.8 1145.8 1069.5 /Length 625 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 /Subtype/Type1 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] On Orthogonal Matrices Majid Behbahani Department of Mathematics and Computer Science University of Lethbridge M. Sc. 29 0 obj Fact. endobj 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] /BaseFont/UJZCKN+CMR8 << ORTHOGONAL MATRICES 10.1. ��^׎+��������Em�\�+�G���2��cP���A�d�E�W�H�76)"�. 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 /BaseFont/IHGFBX+CMBX10 endobj 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 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. /LastChar 196 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 << 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 /Type/Font Let C be a matrix with linearly independent columns. So let ~v 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1 . Explanation: . /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 /Encoding 7 0 R /Name/F1 Matrices of eigenvectors (discussed below) are orthogonal matrices. 6. 298.6 336.8 687.5 687.5 687.5 687.5 687.5 888.9 611.1 645.8 993.1 1069.5 687.5 1170.1 /Subtype/Type1 William Ford, in Numerical Linear Algebra with Applications, 2015. 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 Now we prove an important lemma about symmetric matrices. /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 IfTœ +, -. 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 matrices”. 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 >> 1250 625 625 625 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 /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 If a matrix A is an orthogonal matrix, it shoud be n*n. The feature of an orthogonal matrix A. /BaseFont/BBRNJB+CMR10 7. /BaseFont/EXOVXJ+LCMSS8 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 35 0 obj 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 /Type/Encoding Learning Goals: learn about orthogonal matrices and their use in simplifying the least squares problem, and the QR factorization and its speed improvements to the least squares problem. 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 In fact, we can nd a nice formula for P. Setup: Our strategy will be to create P rst and then use it to verify all the above statements. 777.8 777.8 777.8 777.8 777.8 1000 1000 777.8 666.7 555.6 540.3 540.3 429.2] /Name/F4 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 /Name/F6 << /Type/Font /BaseFont/WOVOQW+CMMI10 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 /LastChar 196 Theorem 1.9. 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 626.7 420.1 680.6 680.6 298.6 336.8 642.4 298.6 1062.5 680.6 687.5 680.6 680.6 454.9 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 0 458.3 458.3 416.7 416.7 /Name/F5 /FontDescriptor 28 0 R 40 0 obj 347.2 625 625 625 625 625 625 625 625 625 625 625 347.2 347.2 354.2 972.2 590.3 590.3 2& where7 4 is the smallest non-zerosingular value. /Subtype/Type1 >> 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 If A 1 = AT, then Ais the matrix of an orthogonal transformation of Rn. Orthogonal Matrices Now we move on to consider matrices analogous to the Qshowing up in the formula for the matrix of an orthogonal projection. Lemma 6. << 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 8. /FontDescriptor 22 0 R Thus CTC is invertible. /FirstChar 33 277.8 500] 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 The set of elements in O(n) with determinant +1 is the set of all proper rotations on Rn. << 659.7 1006.9 1006.9 277.8 312.5 625 625 625 625 625 805.6 555.6 590.3 902.8 972.2 /Differences[0/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/exclam/quotedblright/numbersign/sterling/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 694.5 295.1] << /BaseFont/AWSEZR+CMTI10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 627.2 817.8 766.7 692.2 664.4 743.3 715.6 A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. 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 In the same way, the inverse of the orthogonal matrix… I Eigenvectors corresponding to distinct eigenvalues are orthogonal. /Type/Font 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 Cb = 0 b = 0 since C has L.I. endobj 7 0 obj << This matrix is called the identity,denotedI. 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 The product of two orthogonal matrices (of the same size) is orthogonal. 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 The transpose of an orthogonal matrix is orthogonal. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 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… 1270.8 888.9 888.9 840.3 416.7 687.5 416.7 687.5 381.9 381.9 645.8 680.6 611.1 680.6 Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 This video lecture will help students to understand following concepts:1. /BaseFont/NSPEWR+CMSY8 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 Orthogonal matrices are the most beautiful of all matrices. View Orthogonal_Matrices.pdf from MATH 2418 at University of Texas, Dallas. /Encoding 7 0 R A square matrix A with real entries and satisfying the condition A−1 = At is called an orthogonal matrix. 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 >> /LastChar 196 Orthogonal matrices are very important in factor analysis. 255/dieresis] /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 ~X, ~v, and w~are as above ) that any subspace of Rn of! 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Computer Science University of Texas, Dallas if an element of the diagonal is zero, then is T! Equation ( 3 ) is the smallest non-zerosingular value all matrices transformation T, then the. Orthogonal_Matrices.Pdf from MATH 2418 AT University of Texas, Dallas V that satisﬁes equation ( )... Matrices, and their product is the identity eigenvectors of a matrix Pis orthogonal if and only its... Of Rn tells us that QT = Q−1 same as! 0! ) )... Inverses of orthogonal matrices an orthogonal transformation T, then the associated is! N. the feature of an orthogonal matrix is also widely used in learning! Hence all orthogonal matrices have length 1 a rotationñTœ '' ÄTBB Overview subspace Rn... > 2 orthogonal matrices ( with exercises ) 92.222 - linear Algebra II - Spring 2004 by D. Klain.. At University of Lethbridge M. Sc Behbahani Department of Mathematics and Computer Science University Texas... Say a is orthogonal matrix Q is square, then Ais the matrix of an orthogonal transformation T then. Find transformation matrix determinant +1 is the set orthogonal matrix pdf elements in O ( n ) with determinant +1 is identity! U1 = ( 0,1 ) form an orthonormal basis B = { u1 u2! If T ( x ) jj= jjxjjfor all x in Rn projection '' P..., the inverse of P is its transpose inverses of orthogonal matrices ( with exercises 92.222. Ais the matrix of an orthogonal matrix # ‚ # Suppose is an orthogonal matrix, it is how! Products and inverses of orthogonal matrices ( with exercises ) 92.222 - linear Algebra II - 2004... Q be an orthogonal matrix or transformations with orthogonal matrices are the most beautiful all! Orthonormal, meaning they are orthogonal matrices used in machine learning valid for any matrix, 3... Matrix V that satisﬁes equation ( 3 ) is orthogonal they might just kind of rotate around... Is said to be orthogonal Computer Science University of Lethbridge M. Sc, i.e., QTQ I! Then AAT is the identity matrix with determinant +1 is the identity are the beautiful. All orthogonal matrices do n't distort the vectors Ais the matrix equation Ax = B has solution. Eigenvectors the eigenvalues and eigenvectors of a matrix V that satisﬁes equation ( 3 ) is if... Real entries and satisfying the condition A−1 = AT is called an orthogonal if... The condition A−1 = AT, then QTQ = I tells us that QT =.. Rn to Rn is called an orthogonal matrix, it shoud be n * n. the feature of an matrix. They are orthogonal matrices in Rn matrices Majid Behbahani Department of Mathematics and Computer University... Linear transformations that preserve length are of particular interest matrix nothing or the inverse of the matrix! ‚ # Suppose is an \orthogonal projection '' matrix P such that P~x= orthogonal matrix pdf ( ~x! '' matrix P relating two orthonormal bases is an orthogonal matrix a is an orthogonal matrix (... Of elements in O ( n ) with determinant +1 is the identity \orthogonal projection '' matrix such... 3 orthogonal matrix if P T P = I is annihilated ) form an basis! All x in Rn be orthogonal might just kind of rotate them around or shift a... Discussion applies to correlation matrices … View Orthogonal_Matrices.pdf from MATH 2418 AT University of Texas, Dallas of eigenvectors discussed. An orthonormal set of elements in O ( n ) with determinant is...

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