601-s-p251ff - 5.1 Introduction 251 EXERCISES 5.1.1 Find the eigenvalues and eigenvectors of the matrix A ="4 Verify that the trace equals the

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Unformatted text preview: 5.1 Introduction 251 EXERCISES 5.1.1 Find the eigenvalues and eigenvectors of the matrix A = "4]. Verify that the trace equals the sum of the eigenvalues, and the determinant equals their product. 5.1.2 With the same matrix A, solve the differential equation du/dt = Au, uo = [2]. What are the two pure exponential solutions? 252 5.1 .3 5.1 .4 5.1 .5 5.1.6 5.1 .7 5.1 .8 5.1 .9 5 Eigenvalues and Eigenvectors Suppose we shift the preceding 71 by subtracting 71: —6 —1 B=A—7I= . What are the eigenvalues and eigenvectors of B, and how are they related to those of A? Solve du/dt = Pu when P is a projection: du l i , 5 E = ;]u wrth u0 = The column space component of no increases exponentially while the nullspace com- ponent stays fixed. Find the eigenvalues and eigenvectors of 3 4 2 0 0 2 A= 012 and B: 0 2 0. O O 0 2 0 0 Check that A, + A; + 2.3 equals the trace and 2.11213 equals the determinant. Give an example to show that the eigenvalues can be changed when a multiple of one row is subtracted from another. Suppose that ,l is an eigenvalue of A, and x is its eigenvector: Ax = Ax. (a) Show that this same x is an eigenvector of B = A — 71, and find the eigenvalue. This should confirm Exercise 5.1.3. (b) Assuming 1. ¢ 0, show that x is also an eigenvector of A"'—and find the eigenvalue. Show that the determinant equals the product of the eigenvalues by imagining that the characteristic polynomial is factored into det(A - 11) = (11 - M12 - l) ' ' ' (1.. - l). (15) and making a clever choice of A. Show that the trace equals the sum of the eigenvalues, in two steps. First, find the coefficient oft—A)” 1 on the right side of (15). Next, look for all the terms in an — '1 “12 ' ' ' aln - a a — A ~' a dam — 21) = det ?‘ 22 2" anl an: arm — ’1 which involve ( —A)"' 1. Explain why they all come from the product down the main diagonal, and find the coefficient of (—,l)"'1 on the left side of (15). Compare. 5.1.10 5.1.11 5.1 .1 2 5.1 .13 5.1 .14 5.1 .15 5.1 .16 5.1 .1 7 5.1 .18 5.1 Introduction 253 ’ (a) Construct 2 by 2 matrices such that the eigenvalues of AB are not the products of the eigenvalues of A and B, and the eigenvalues of A + B are not the sums of the individual eigenvalues. (b) Verify however that the sum of the eigenvalues of A + B equals the sum of all the individual eigenvalues of A and B, and similarly for products. Why is this true? Prove that A and AT have the same eigenvalues, by comparing their characteristic polynomials. Find the eigenvalues and eigenvectors of A = [i -3] and A = [Z If B has eigenvalues I, 2, 3 and C has eigenvalues 4, 5, 6, and D has eigenvalues 7, 8, 9, what are the eigenvalues of the 6 by 6 matrix A = [g 5]? Find the rank and all four eigenvalues for both the matrix of ones and the checker- board matrix: 0101 10 10 and C=0 101. l 010 HF‘D‘H _)-‘D_U— I‘D-‘HO—D HD-‘O—tD—fi Which eigenvectors correspond to nonzero eigenvalues? What are the rank and eigenvalues when A and C in the previous exercise are n by n? Remember that the eigenvalue ,1 = 0 is repeated n — r times. If A is the 4 by 4 matrix of ones, find the eigenvalues and the determinant of A — I (compare Ex. 4.3.10). Choose the third row of the “companion matrix” 0 l O A = O 0 1 so that its characteristic polynomial |A — AI| is —/l3 + 4/12 + 5.1 + 6. Suppose the matrix A has eigenvalues 0, I, 2 with eigenvectors no, 01, 02. Describe the nullspace and the column space. Solve the equation Ax = 01 + v2. Show that Ax = 00 has no solution. 4.4.1 6 4.4.17 5.1.2 5.1.4 5.1.7 5.1.8 5.1.9 5.1.12 5.1.14 5.1.15 5.1.18 5.2.1 5.2.4 5.25 5.2.6 5.2.8 5.2.9 5.2.12 5.2.14 Solutions to Selected Exercises 483 The power of 'P are all permutation matrices so eventually one must be repeated. If P' = P‘ then'P'"s =-I. By formula (6), det A s 5!: 120. Or by det A = volume, det A 3 (J5)? Or by pivots,detAsl x 2x4x 8 x16. CHAPTER 5 _ 1 2r 1 31__ 1 2r_ 1 3: u—c,[_l]e +c2[_2]e —6[_1 e 6 _2 e. ‘ 1 1 , u—[_l]+4[l]e. . Ax =).x=>(A — 71)x = (,1— 7)x; Ax=lx => x=lA"x => A"x= (1/11)x. Choosel=0. The coefficient is A, + - ' - + A". In detU. —- 11), a term which includes an ofl—diagonal aij excludes both a“ — 1. and a j,- — 1. Therefore such a term doesn’t involve (—l)"" ‘. The coefficient of (—).)"‘I must come from the main diagonal and it is al , + ..._|_a’m=)bl 2 l 11=5Niz=—5ax1=[1]ax2=[ é];11=a+b112=a*b:x1=[1]. x _ 1 2— -1' rank(A)= 1;). = 0, 0, 0, 4; x4 =(l,1,l,l);rank(B)= 2; A = 0, 0, 2, —2; x3 =(l,1,1, 1), x4 = (1, —-l, 1, ~1). rank(A) = l, 2. = 0,. . . , 0, n; rank(C) = 2, A = 0, . . . , n/2, —n/2. The nullspace is spanned by 00; x = c0110 + v, + %1)2; 00, 1),, oz are independent (the eigenvalues are distinct), => v0 is not in the column space which is spanned by vl and v2. 112011".112011‘l [1 Alla oil: a] ’[o —2][o ollo —2] - It has distinct eigenvalues 1, 2, 7; A = diag(l, 2, 7). Al and A3 cannot be diagonalized. (a) 1. = 1 or —1 (b) trace = 0; determinant = —l (c) (8, —3). (a) Au = uvTu = (vTu)u => A = vTu. (b) All other eigenvalues are zero because dim JV(A)= n - l. trace(AB) = trace(BA) = aq + bs + cr + dt =» trace(AB — BA) = 0 => AB — BA = I is impossible. 1 1 9 o 1 1-1 2 1 . = - -4. 5313 A [1 —1][o 1][1 —l] ’[1 2} det A = det(S/\S") = det Sdet A det S"I = det A = [Mn-1.". F; T; T. ...
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This note was uploaded on 11/08/2011 for the course MATH 601 taught by Professor Osu during the Fall '11 term at Cornell University (Engineering School).

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601-s-p251ff - 5.1 Introduction 251 EXERCISES 5.1.1 Find the eigenvalues and eigenvectors of the matrix A ="4 Verify that the trace equals the

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