/Name/F3 /Name/F3 matrices”. 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 /Type/Font 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 627.2 817.8 766.7 692.2 664.4 743.3 715.6 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 625 352.4 625 347.2 347.2 590.3 625 555.6 625 555.6 381.9 625 625 277.8 312.5 590.3 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 /BaseFont/QQXJAX+CMMI8 /Type/Font /LastChar 196 The collection of the orthogonal matrix of order n x n, in a group, is called an orthogonal group and is denoted by ‘O’. 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 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 << /LastChar 196 << /FontDescriptor 34 0 R The product AB of two orthogonal n £ n matrices A and B is orthogonal. /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 0 0 0 0 722.2 555.6 777.8 666.7 444.4 666.7 777.8 777.8 777.8 777.8 222.2 388.9 777.8 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 458.3 381.9 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 687.5 381.9 What is Orthogonal Matrix? if det , then the mapping is a rotationñTœ" ÄTBB /BaseFont/IHGFBX+CMBX10 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 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 /Subtype/Type1 /Subtype/Type1 /Type/Font 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 >> << /FontDescriptor 28 0 R They might just kind of rotate them around or shift them a little bit, but it doesn't change the angles between them. /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 Lemma 6. 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] /Name/F10 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 0 826.4 295.1 826.4 531.3 826.4 381.9 392.4 1069.5 649.3 649.3 916.7 888.9 902.8 878.5 979.2 854.2 816 916.7 899.3 IfTœ +, -. << endobj /Type/Encoding /BaseFont/CXMPOE+CMSY10 /FirstChar 33 If an element of the diagonal is zero, then the associated axis is annihilated. 826.4 295.1 531.3] Thus, if matrix A is orthogonal, then is A T is also an orthogonal matrix. 2& where7 4 is the smallest non-zerosingular value. Let us now rotate u1 and u2 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 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 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 /BaseFont/AWSEZR+CMTI10 >> 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 0 663.6 885.4 826.4 736.8 Proof. 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 /FontDescriptor 12 0 R /FirstChar 33 26 0 obj 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 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. 9. 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 Matrices of eigenvectors (discussed below) are orthogonal matrices. This is valid for any matrix, regardless of the shape or rank. /BaseFont/AUVZST+LCMSSB8 /Subtype/Type1 Hence all orthogonal matrices must have a determinant of ±1. endobj columns. 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. 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 << 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 0 830.6 670.8 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 /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 /Widths[306.7 514.4 817.8 769.1 817.8 766.7 306.7 408.9 408.9 511.1 766.7 306.7 357.8 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 endobj endobj >> 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 /LastChar 127 Every n nsymmetric matrix has an orthonormal set of neigenvectors. The set of elements in O(n) with determinant +1 is the set of all proper rotations on Rn. A great example is projecting onto a subspace. << 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 << 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 0 826.4 295.1 826.4 531.3 826.4 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 << The di erence now is that while Qfrom before was not necessarily a square matrix, here we consider ones which are square. Figure 4 illustrates property (a). >> 812.5 916.7 899.3 993.1 1069.5 993.1 1069.5 0 0 993.1 802.1 722.2 722.2 1104.2 1104.2 0 0 1 0 1 0 For example, if Q = 1 0 then QT = 0 0 1 . 694.5 295.1] << 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 /FirstChar 0 >> /Type/Font /FirstChar 33 The transpose of an orthogonal matrix is orthogonal. A matrix V that satisﬁes equation (3) is said to be orthogonal. /FontDescriptor 31 0 R I To show these two properties, we need to consider complex matrices of type A 2Cn n, where C is the set of Orthogonal Matrices. 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 This discussion applies to correlation matrices … stream endobj Exercise 3.6 What is the count of arithmetic ﬂoating point operations for evaluating a matrix vector product with an n×n 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 >> /FontDescriptor 12 0 R I Eigenvectors corresponding to distinct eigenvalues are orthogonal. 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 0 298.6 336.8 687.5 687.5 687.5 687.5 687.5 888.9 611.1 645.8 993.1 1069.5 687.5 1170.1 /Type/Font )��R\$���_W?՛����i�ڷ}xl����ڮ�оo��֏諭k6��v���. Recall that Q is an orthogonal matrix if it satisfies QT = Q−1 . So orthogonal vectors make things much easier. /BaseFont/NSPEWR+CMSY8 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 Products and inverses of orthogonal matrices a. 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. Note that for a full rank square matrix, !3) is the same as !0!). << 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 /Filter[/FlateDecode] 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 The product of two orthogonal matrices (of the same size) is orthogonal. 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 Orthogonal matrices are the most beautiful of all matrices. /LastChar 196 /Subtype/Type1 The set of vectors that are annihilated by the matrix form a vector space [prove], which is called the row nullspace,orsimplythenullspace of the matrix. The orthonormal set can be obtained by scaling all vectors in the orthogonal set of Lemma 5 to have length 1. 3. endobj Let C be a matrix with linearly independent columns. 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 /Type/Font A square orthonormal matrix Q is called an orthogonal matrix. /Type/Font /LastChar 196 endobj 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 /Encoding 7 0 R Orthogonal matrix • 2D example: rotation matrix nothing. orthogonal matrix is a square matrix with orthonormal columns. endobj %PDF-1.2 Now we prove an important lemma about symmetric matrices. /Filter[/FlateDecode] 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 %PDF-1.2 /FirstChar 33 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 16 0 obj /LastChar 196 /Encoding 7 0 R 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 Explanation: . 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 /Name/F4 /Name/F5 /BaseFont/MITRMO+MSBM10 /BaseFont/WOVOQW+CMMI10 6. Alternatively, a matrix is orthogonal if and only if its columns are orthonormal, meaning they are orthogonal and of unit length. endobj Example 10.1.1. /FirstChar 33 De nitions and Theorems from 5.3 Orthogonal Transformations and Matrices, the Transpose of a Matrix De nition 1. Then to summarize, Theorem. /FontDescriptor 18 0 R Thus, a matrix is orthogonal … /Type/Encoding 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 694.5 295.1] In this tutorial, we will dicuss what it is and how to create a random orthogonal matrix with pyhton. 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 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. 5) Norm of the pseudo-inverse matrix The norm of the pseudo-inverse of a (×*matrix is:!3=.-3,#!3)=! 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 View Orthogonal_Matrices.pdf from MATH 2418 at University of Texas, Dallas. 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 19 0 obj 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 9 0 R /Encoding 7 0 R /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 endobj 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 /Name/F4 << 625 1062.5 1201.4 972.2 277.8 625] 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 \$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�>�� 10 0 obj 7. 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