Proof. 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 20 0 obj There is an \orthogonal projection" matrix P such that P~x= ~v(if ~x, ~v, and w~are as above). Note. The following are equivalent characterizations of an orthogonal matrix Q: 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 /Type/Encoding 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 We prove that eigenvalues of orthogonal matrices have length 1. stream 826.4 295.1 531.3] /LastChar 196 Orthogonal matrices are very important in factor analysis. /Subtype/Type1 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 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 /FirstChar 33 >> �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���/��>� columns. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 A matrix P is orthogonal if P T P = I, or the inverse of P is its transpose. If a matrix A is an orthogonal matrix, it shoud be n*n. The feature of an orthogonal matrix A. i.e. /FirstChar 33 The transpose of an orthogonal matrix is orthogonal. Orthogonal Matrices and QR. 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 >> /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 That is, T is orthogonal if jjT(x)jj= jjxjjfor all x in Rn. Orthogonal matrix • 2D example: rotation matrix nothing. 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 /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 /Subtype/Type1 /FontDescriptor 12 0 R 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 The vectors u1 =(1,0) and u2 =(0,1) form an orthonormal basis B = {u1,u2}. << /Name/F3 endobj /Type/Encoding 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 38 0 obj 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 << /BaseFont/CYTIPA+CMEX10 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. 625 352.4 625 347.2 347.2 590.3 625 555.6 625 555.6 381.9 625 625 277.8 312.5 590.3 The most desirable class of matrices … 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 The product of two orthogonal matrices (of the same size) is orthogonal. 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 /BaseFont/EXOVXJ+LCMSS8 /LastChar 196 << 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 >> This discussion applies to correlation matrices … /Length 2119 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 0 777.8 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 /Length 625 << Thus, a matrix is orthogonal … /Name/F9 << In this tutorial, we will dicuss what it is and how to create a random orthogonal matrix with pyhton. 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 /LastChar 196 /LastChar 196 endobj /Widths[354.2 625 1041.7 625 1041.7 937.5 347.2 486.1 486.1 625 972.2 347.2 416.7 This video lecture will help students to understand following concepts:1. 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 0 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 Explanation: . /BaseFont/NSPEWR+CMSY8 /FontDescriptor 34 0 R 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 Eigenvalues and Eigenvectors The eigenvalues and eigenvectors of a matrix play an important part in multivariate analysis. William Ford, in Numerical Linear Algebra with Applications, 2015. 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 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 Every n nsymmetric matrix has an orthonormal set of neigenvectors. 5) Norm of the pseudo-inverse matrix The norm of the pseudo-inverse of a (×*matrix is:!3=.-3,#!3)=! 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 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 /Subtype/Type1 9. 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. 255/dieresis] /FirstChar 33 361.1 635.4 927.1 777.8 1128.5 899.3 1059 864.6 1059 897.6 763.9 982.6 894.1 888.9 >> /LastChar 196 The orthonormal set can be obtained by scaling all vectors in the orthogonal set of Lemma 5 to have length 1. 1062.5 826.4] /Encoding 7 0 R 3gis thus an orthogonal set of eigenvectors of A. Corollary 1. Let A be an n nsymmetric matrix. x��Z[�ܶ~���`1�_��E��m������7ί�!)J���ٛ�eG�y.�΅R��B! /FontDescriptor 9 0 R Let C be a matrix with linearly independent columns. 277.8 500] endobj A great example is projecting onto a subspace. Products and inverses of orthogonal matrices a. /Subtype/Type1 Suppose CTCb = 0 for some b. bTCTCb = (Cb)TCb = (Cb) •(Cb) = Cb 2 = 0. >> Hence all orthogonal matrices must have a determinant of ±1. 381.9 392.4 1069.5 649.3 649.3 916.7 888.9 902.8 878.5 979.2 854.2 816 916.7 899.3 >> 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 /FontDescriptor 25 0 R 1062.5 826.4] 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 /Filter[/FlateDecode] 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. /Type/Font Lemma 6. 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 /Type/Font 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 As we will now prove, this set is a subgroup of O(n) and it is called the special orthogonal group, denoted SO(n). endobj /FirstChar 33 255/dieresis] if det , then the mapping is a rotationñTœ" ÄTBB If Q is square, then QTQ = I tells us that QT = Q−1. /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 0 826.4 295.1 826.4 531.3 826.4 endobj /Type/Font Overview. 16 0 obj Is the product of k > 2 orthogonal matrices an orthogonal matrix? 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 )��R$���_W?՛����i�ڷ}xl����ڮ�оo��֏諭k6��v���. /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 /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 Rn is called an orthogonal matrix of Lethbridge M. 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