orthogonal matrix pdf

/Type/Font 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 /Name/F4 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 << 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 << Note. The di erence now is that while Qfrom before was not necessarily a square matrix, here we consider ones which are square. 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… 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 9. So orthogonal vectors make things much easier. /FontDescriptor 31 0 R /BaseFont/AUVZST+LCMSSB8 /Name/F3 �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���/��>� /LastChar 196 /FirstChar 33 stream /FontDescriptor 12 0 R /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 277.8 972.2 625 625 625 625 416.7 479.2 451.4 625 555.6 833.3 555.6 555.6 538.2 625 columns. /Encoding 20 0 R /FirstChar 33 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 19 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 Consider the euclidean space R2 with the euclidean inner product. 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 There is an \orthogonal projection" matrix P such that P~x= ~v(if ~x, ~v, and w~are as above). 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 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. /BaseFont/WOVOQW+CMMI10 << Example 10.1.1. 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 777.8 1000 1000 1000 1000 1000 1000 777.8 777.8 555.6 722.2 666.7 722.2 722.2 666.7 /BaseFont/NSPEWR+CMSY8 /FontDescriptor 37 0 R /Type/Font 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 A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. 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. /BaseFont/EXOVXJ+LCMSS8 /Name/F3 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 Cb = 0 b = 0 since C has L.I. /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 Orthogonal matrices are very important in factor analysis. If an element of the diagonal is zero, then the associated axis is annihilated. /Length 625 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 23 0 obj What is Orthogonal Matrix? ORTHOGONAL MATRICES 10.1. Orthogonal Transformations and Matrices Linear transformations that preserve length are of particular interest. 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 In this tutorial, we will dicuss what it is and how to create a random orthogonal matrix with pyhton. 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 Theorem 1.9. /LastChar 196 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 575 1041.7 1169.4 894.4 319.4 575] << 7 0 obj 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 /FontDescriptor 22 0 R 32 0 obj >> FUNDAMENTALS OF LINEAR ALGEBRA James B. Carrell carrell@math.ubc.ca (July, 2005) 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 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 /Subtype/Type1 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 /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 >> endobj /FirstChar 33 Introduction Definition. This discussion applies to correlation matrices … 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 0 458.3 458.3 416.7 416.7 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1 . 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 Hence all orthogonal matrices must have a determinant of ±1. 3. << 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 If A 1 = AT, then Ais the matrix of an orthogonal transformation of Rn. 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 Products and inverses of orthogonal matrices a. 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 40 0 obj The vectors u1 =(1,0) and u2 =(0,1) form an orthonormal basis B = {u1,u2}. 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 endobj endobj They might just kind of rotate them around or shift them a little bit, but it doesn't change the angles between them. /Type/Font /LastChar 196 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. Orthogonal matrices are the most beautiful of all matrices. 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 /Subtype/Type1 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 endobj 3gis thus an orthogonal set of eigenvectors of A. Corollary 1. /LastChar 196 /Type/Font 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 Recall that Q is an orthogonal matrix if it satisfies QT = Q−1 . If a matrix A is an orthogonal matrix, it shoud be n*n. The feature of an orthogonal matrix A. /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 I Eigenvectors corresponding to distinct eigenvalues are orthogonal. >> That is, T is orthogonal if jjT(x)jj= jjxjjfor all x in Rn. << 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.

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