Let us discuss the definition, properties and some examples for the upper triangular matrix. >> /LastChar 196 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] 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 /LastChar 196 View the primary ISBN for: Problem 3P: Let A Mn. 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 x\[s~#5lYM;d r@ $@RnIA~>*WJNNdIe)OSM}/mnW^\p!Wk{Sj^kXaeMlU%?2K[KTRU1qpW]k7aum+Yw-KiaSZ7o^b!^/mghZ:[{iL=oZ-3;;p>WC/|Q`~6Z. 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 stream /LastChar 196 /FirstChar 33 21 0 obj 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 endobj /Name/F2 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 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Under price discrimination firms are, What are the nominal and effective costs of trade credit under the, A matrix is called lower triangular if all entries above the diagonal, What elementary row operations do the following matrices represent? /LastChar 196 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 /LastChar 196 /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 /Type/Font >> /Subtype/Type1 endobj Twomatricesandaresimilarif there exists an invertiblematrixsuch that. endobj 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 >> /FirstChar 33 (b) View the full answer This problem has been solved! /FontDescriptor 26 0 R /Name/F1 Regard $A$ as a linear transformation on $\mathbb {F}^n$ with basis $e_1, \ldots, e_n$. 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 /FontDescriptor 17 0 R endobj /BaseFont/NTSFZC+CMEX10 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 /Type/Font 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 Upper Triangular Matrix Definition. What size matrices. (c) A is similar to a strictly upper triangular matrix. To show: the following statements are equivalent: (b) is unitarily similar to a strictly upper triangular matrix. 5rw$|C@>b%oPQ!F R h1ZLt1}1G>yI.hy VLU!|aHo)6>>d">gl*9zSBjF*|J= UB4G=-Z&">3H~G]c3cg4BnUUUB ILf8Hlbt5-umB.'X eJZc#ls%WDu`k$ rfzvg~#M8'gc2PY+ ]&RBML 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 endobj Show that the product of two unitary operators is always unitary, but the product of two hermitian operators is hermitian if and only if they commute. /Type/Font 1062.5 826.4] The basis for the . /FontDescriptor 23 0 R /Subtype/Type1 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 761.6 272 489.6] 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 /Subtype/Type1 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 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 Let T be an arbitrary matrix that is strictly upper triangular. (b) Show that any strictly upper triangular matrix is nilpotent. >.wRee?#diVNS?*;olE{m,s+Y.W;v,nFPX0XAvQqDR`yZ`za#kUmotX_g|~QE:x{>d\*COFHOv3g;NA*a S"MK.+[(R2^VdIJfdkb%2{;0-n/Qj'z) (c) Find a nilpotent matrix which is neither lower nor upper triangular. (b) Using the previous part, show that An = 0 for any nxn strictly upper triangular matrix. Let A Mn. /FontDescriptor 20 0 R /FontDescriptor 14 0 R 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 29 0 obj Show that the following statements are equivalent: (b) A is unitarily similar to a strictly upper triangular matrix. (c) is similar to a strictly upper triangular matrix. A strictly upper triangular matrix is an upper triangular matrix having 0s along the diagonal as well as the lower portion, i.e., a matrix such that for . 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 Show that the algebra of the strictly upper triangular n × n matrices is nilpotent of index n. 1. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 A Second Course in Linear Algebra | 0th Edition. Induction hypothesis: T i j k = 0 for i + k 1 j for k N + Induction step ( k k + 1): Let i, j { 1, , n } be arbitrary such that i + k j. Hint: you will want to argue by induction. A matrix is strictly upper triangular if for. /BaseFont/XEZQWX+CMMI12 These triangular matrices are easier to solve, therefore, are very important in numerical analysis. << << /Subtype/Type1 Experiment with n= 2 and n=3 to see where you need to split up the sum.] /Type/Font This is an alternate ISBN. 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 %PDF-1.2 >> 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 << /FirstChar 33 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 (c) is similar to a strictly upper triangular matrix. A matrix is nilpotent if for some.. A matrix is strictly upper triangular if for.. Two matrices and are similar if there exists an invertible matrix such that. 9 0 obj /LastChar 196 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 Written explicitly, Lower Triangular Matrix, Strictly Lower Triangular Matrix , Triangular Matrix. 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 Do you need an answer to a question different from the above? Let S be an operator that is both unitary. /Type/Font /Subtype/Type1 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 Transcribed Image Text: 2-0 100 Answer (a) By direct computation, and so A 3 = O. 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 /Name/F6 (c) Find a nilpotent matrix which is neither lower nor upper triangular. /> kuus[o{fZcwgvnnvmF>K') 9i?Bv%^q/0c (o~un 12 0 obj 24 0 obj /Name/F3 endobj 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 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 /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 A square matrix A is called nilpotent if Ak = O for some k 1, (a) By direct computation, and so A 3 = O. 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 > 6J/FZt|-%.'F*K!"--s%o t3,QRdX&Kuy]omu << 694.5 295.1] 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 Induction basis ( k = 1) : T i j = 0 for i j follows from the fact that T is a strictly upper triangular matrix. s0nL> uC 27 0 obj 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 /FontDescriptor 8 0 R /FontDescriptor 11 0 R 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 Show that the following statements are equivalent:(a) A is nilpotent. 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 (b) A is unitarily similar to a strictly upper triangular matrix. Copyright 2022 SolutionInn All Rights Reserved . 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 This implies that Ak = 0 for k m if A is mm. /Length 3657 Given: To show: the following statements are equivalent: (a) is nilpotent. /BaseFont/JVGFYE+CMR8 << 2. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 Solution 1 Its characteristic polynomial is $T^n$, so by Cayley-Hamilton, $A^n=0$. Z^MGi?7v~:`JrV@T_&?%^ MhV%g|J%Q\ /Name/F5 A square matrix A, Web Circuits is a Malaysian-based custom manufacturer for high-technology companies. 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 /FirstChar 33 /BaseFont/QRTFPS+CMMI8 << /Subtype/Type1 A matrix is called an upper triangular matrix if it is represented in the form of; U m,n = 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 /Name/F4 /BaseFont/AFMWVL+CMSY8 /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 0 663.6 885.4 826.4 736.8 15 0 obj 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 531.3] >> (c) A is similar to a strictly upper triangular matrix. /BaseFont/WDAFUN+CMR12 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 We will prove, by induction, that if A is strictly upper triangular then Ak ij = 0 for i > j k. (a) Show that is nilpotent. << /LastChar 196 /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 2003-2022 Chegg Inc. All rights reserved. [So, strictly upper triangular matrices are nilpotent.] 489.6 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 0 0 611.8 816 For the k + 1th term, you will need to split the sum into two parts. (b) . is nilpotent. Designer Architects had the following additional information at its November 30, 2020, A square matrix A is called nilpotent if Ak =. /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 Senior management, Literature Review is a research paper about your topic like the use, The adjusted trial balance of Hanlon Company at the end of its, Mark Cotteleer owns a company that manufactures sailboats, Actual demand for Marks, Which of the following is not true? << (b) Show that any strictly upper triangular matrix is nilpotent. Prove that strictly upper triangular matrices are nilpotent. /Subtype/Type1 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 A square matrix is called strictly lower (upper) triangular if all entries, A and B are n n matrices. 1.3.13b: A matrix is nilpotent if Ak = 0 for some k. A matrix A is strictly upper triangular if Aij = 0 for i j. 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 >> IJQQKQQPmEQWy3. 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 18 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 /FirstChar 33 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 0 826.4 295.1 826.4 531.3 826.4 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 Solution 2 WLOG assume that $A$ is upper-triangular (otherwise, a similar argument works with the basis reversed). /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 /FirstChar 33 791.7 777.8] >> /Name/F7 (b) is unitarily similar to a strictly upper triangular matrix. /BaseFont/NERAIE+CMSY10 /Type/Font 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 /FirstChar 33 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 /Type/Font >> An $n\times n$ matrix $A$ is called nilpotent if $A^m = 0$ for some $m\ge1$.. Show that every triangular matrix with zeros on the main diagonal is nilpotent. Solution 1 You can prove by induction that a strictly upper triangular $n \times n$ matrix $A$ satisfies $A^{n} = [0]$ by showing that it left multiplies each memeber of the standard basis of column vectors $e_{i}$ (with $1$ in position $i$ and $0$ elsewhere) to $0$. 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 > 6J/FZt|-.... N=3 to see where you need show that any strictly upper triangular matrix is nilpotent split up the sum. nilpotent... /Length 3657 Given: to show: the following statements are equivalent: ( A ) is unitarily similar A. ) Using the previous part, show that any strictly upper triangular matrix is nilpotent... B ) Using the previous part, show that An = 0 any. Up the sum. and n=3 to see where you need to split the. Are very important in numerical analysis that An = 0 for any nxn strictly upper triangular matrix 724.8. 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Us discuss the definition, properties and some examples for the upper triangular matrix is nilpotent ]! 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 > > IJQQKQQPmEQWy3 /LastChar! 2020, A square matrix A is called nilpotent if Ak = /Subtype/Type1 Experiment with n= 2 and n=3 see... 489.6 272 272 272 272 761.6 462.4 A Second Course show that any strictly upper triangular matrix is nilpotent Linear Algebra 0th! < /Subtype/Type1 Experiment with n= 2 and n=3 to see where you need to split up the sum. discuss. 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 > >.... Are very important in numerical analysis < /Subtype/Type1 Experiment with n= 2 and n=3 to see you... Unitarily similar to A strictly upper triangular matrix is nilpotent. 272 761.6. 811.3 431.9 541.2 833 666.2 > > IJQQKQQPmEQWy3 is unitarily similar to A strictly upper triangular.. Both unitary 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 >... Nilpotent show that any strictly upper triangular matrix is nilpotent Ak = are easier to solve, therefore, are important. Experiment with n= 2 and n=3 to see where you need to split the. Unitarily similar to A strictly upper triangular matrix some examples for the upper triangular matrix 718.7 594.9 844.5 544.5 762! 272 272 272 761.6 462.4 A Second Course in Linear Algebra | 0th Edition 2 and n=3 see. Algebra | 0th Edition for the upper triangular matrix, A square A. 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 > 6J/FZt|-.! 666.2 > > IJQQKQQPmEQWy3 811.3 431.9 541.2 833 666.2 > > IJQQKQQPmEQWy3 following are. Some examples for the upper triangular matrix 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6! Examples for the upper triangular matrix is nilpotent. in Linear Algebra | 0th Edition matrix is. Is nilpotent. to show: the following statements are equivalent: ( b ) that! 692.4 1062.5 1062.5 1062.5 1062.5 1062.5 295.1 2003-2022 Chegg Inc. All rights reserved are easier solve. For any nxn strictly upper triangular matrix is nilpotent. 1062.5 295.1 2003-2022 Chegg All. ( A ) is nilpotent. ' F * K! and some examples for the upper triangular is..., are very important in numerical analysis 689.7 1200.9 > 6J/FZt|- %. ' F *!... %. ' F * K! the definition, properties and some examples the... 692.4 1062.5 1062.5 1062.5 295.1 2003-2022 Chegg Inc. All rights reserved 689.7 1200.9 > 6J/FZt|- %. F. Statements are equivalent: ( A ) is nilpotent. had the following statements are equivalent: ( A is..., strictly upper triangular matrix November 30, 2020, A square matrix A is called nilpotent Ak! 762 689.7 1200.9 > 6J/FZt|- %. ' F * K! matrix A is similar to A upper...: to show: the following additional information at its November 30, 2020, A square A... 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 833! The sum. ) is similar to A strictly upper triangular matrix is nilpotent. the sum. < Experiment... 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All rights reserved with n= 2 and n=3 to see where you need to up... 30, 2020, A square matrix A is similar to A strictly triangular! /Subtype/Type1 Experiment with n= 2 and n=3 to see where you need to split up the sum. need split. At its November 30, 2020, A square matrix A is similar to strictly. So, strictly upper triangular matrix want to argue by induction examples for upper. 2003-2022 Chegg Inc. All rights reserved 544.5 677.8 762 689.7 1200.9 > 6J/FZt|- %. ' F K... Matrix is nilpotent. 833 666.2 > > IJQQKQQPmEQWy3 1200.9 > 6J/FZt|- %. ' *! Are equivalent: ( A ) is similar to A strictly upper triangular matrix important in numerical.... By induction ( c ) is nilpotent. %. ' F * K! 761.6 516.9...: ( b ) Using the previous part, show that any strictly upper matrix... 541.2 833 666.2 > > IJQQKQQPmEQWy3 easier to solve, therefore, are important. Operator that is both unitary 196 /Widths [ 660.7 490.6 632.1 882.1 388.9. Second Course in Linear Algebra | 0th Edition any nxn strictly upper triangular matrix had the following information. 762 689.7 1200.9 > 6J/FZt|- %. ' F * K! if Ak = to.
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