15. T(x,y)=(x+4y,2a+3y) operator that maps a function into its second derivative. cos,zx are eigenvectors of D2, and find their corre

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1 1. [ 2J X[1j P 21. r q [Suggestion: Work with the standard matrix for the operator.] vector of, and find the corresponding eigenvalue. eigenspace of the linear operator defined by the stated formula. In Exercises (4, confirm by multiplication that x is an eigen- In Exercises 15-16, find the eigenvalues and a basis for each Exercise Set 5.1 r 6. (a) cos,zx are eigenvectors of D2, and find their corre. = 2 2 ; x= 2 (a) ShowthatD2islinear. 4 I P operator that maps a function into its second derivative. I 4 LI (b) Show that if a, is a positive constant, then sin q ax and 2 I I I I I 2 I 18. (Calcuks required) Let D1: C * Cx be the linear operator L 1J L Ij 19. (a) Reflection about the line y = a (e) Shear in the y-direction by a factor k (k ). I,] 6 8 o i I corresponding eigenspaces of the stated matrix operator on R I Refer to the tables in Section 4.9 and use geometric reasoning to 1. I I I 4 f 1 (b) Orthogonal projection onto the az-plane. I 2j & 4j through an angle of I 9 s 6 l (c) Counterclockwise rotation about the positive y-axis o 7 Cd) Dilation with factork (k > I) matrix by inspection. 22. (a) Reflection about the ac-plane. through an angle of iso. (d) Contraction with factork (< k < I). (b) Orthogonal projection onto the y:-plane. In Exercises 1-1, find the characteristic equation of the II (c) Counterclockwise rotation about the positive a-axis 21. (a) Reflection about the av-plane. find the answers. No computations are needed. In each part of Exercises 2122, find the eigenvalues and the (d) Expansion in the y-direction with factor k (k> I). (c) Dilation with factork (k > I). (b) Orthogonal projection onlo the a-axis. (d) Contraction with factork (< k < 1). (c) 1 1 (d) 712, (a) Reflection about the v-axis. - In Exercises find the characteristic equation, the eigen- [I 21 find the answers. No computations are needed. Refer to the tables in Section 4.9 and use geometric reasoning to 5. (a) the eigenvalues, and bases for the eigenspaces of the matrix, corresponding eigenvalues. In each part of Exercises 5-4,, find the characteristic equation, sinh,/x and cosh,jx are eigenvectors of D2, and find their [2 j [ I 2j corresponding eigenspaces of the stated matrix operator on values, and bases for the eigenspaces of the matrix, (b) Rotation about the origin through a positive angle of 18:. in Exercise 17. Show that if w is a positive constant, then 1 4] r 1 In each part of Exercises 19-2(1, find the eigenvalues and the [1 2] (b) [ 2] 4.= 1 2 I ;x= I sponding eigenvalues. j X[1j 17. (Calculus required) Let D2: C(, ) + C(, ) be the L=[ P 11 Ill 16. T(a,y,z)=(2syz,x:, x+y+2:) 15. T(x,y)=(x+4y,2a+y) (c) Rotation about the origin through a positive angle of 9. (c) [ ] (d) { fi (e) Shear in the a-direction by a factor k (k ). Chapter 5 Elgenvalues and Eigenvectors

2 be expressed as 2 (a) = [ ] tr() (a) p() = 22 1)( )2( (b) p() = Use the result of Exercise 28 to prove that if dp 24. Find det() given that has p() as its characteristic poly, = [(a+d)+1,/(a _+4bc] eigenvalues the given matrix. [a (b) = [ 1] are eigenvectors of that correspond, respectively, to the = [ a,] equations for all lines in R2, if any, that are invariant under xi = and x2 2] b r_, R2 inrariant under if x lies on the line when x does. Find 2. Let be a 2 x 2 matrix, and call a line through the origin of. Let be the matrix in Exercise 29. Show that ill, then 4), ) 5.1 Eigenvalues and Eigenvectors 1 (a) That is the size of? (Stated informally, satisfies its characteristic equation. This + 4bc] isi. s cigenvalues? and x is a corresponding eigenvector. values related to the left eigenvectors and their corresponding a corresponding eigenvector, then I/ is an eigenvalue of how are the right eigenvectors and their corresponding eigen-. Prove: If is an eigenvalue of an invertible matrix and xis equation xt = tx for some scalar z. For a given matrix, then (b) Is invertible? result is true as well for n x n matrices.) vectors, which are n x 1 column matrices x that satisfy the has integer eigenvalues. (c) complex conjugate elgenvalues if (a (b) two repeated real eigenvalues if (a (a) two distinct real eigenvalues if (a + 4bc >. + 4bc = + $bc <. for which and T have different eigenspacesl [Hint: Use the result in Exercise to find a 2 x 2 matrix (b) Show that and T need not have the same eigenspaces. equation det(/ =.] Use this result to show that has the same eigenvalues [Hint: Look at the characteristic 8. (a) Prove that if is a square matrix, then and T have = [a + d) ± (a then the solutions of the characteristic equation of are has degree a and that the coefficient of in that polynomial 7. Prove that the characteristic polynomial of ann x n matrix [ dj a b1 (a) (b)1 (c) +2! bases for the eigenspaces of 29. Use the result in Exercise 28 to show that if and then use Exercises and 4 to find the eigenvalues and tnceof. L J ) + det() =, where tr() is the I I =l Prove that the characteristic equation of a 2 x 2 matrix can r2 2 ] Working with Proofs 6. Find the eigenvalues and bases for the eigenspaces of are their corresponding eigenvectors. and xis a corresponding eigenvector. ii Loi L oj eigenvector, then s is an eigenvalue of sit for every scalars for which vector, and s is a scalar, then r I] Eli F fl and xis acorrespondingeigenvector. is an eigenvalue of Lj I II I, I l I 5. Prove: If is an eigenvalue of and x is a corresponding 27. Find a x matrix that has eigenvalues 1, 1, and, and 4. Prove: If is an eigenvalue of, x is a corresponding eigen si called right eigenrectors to distinguish them from left eigen- Lc dj (c) How many eigenspaces does have? 2. Prove: If a, b, c, and d are integers such that a + b = c + d, 26. The eigenvectors that we have been studying are sometimes = Ea 1 answer the question and explain your reasoning. pm) = 2 + c, + c21 = is found to be p() = ( 25. Suppose that the characteristic polynomial of some matrix is the characteristic polynomial of a 2 x 2 matrix, then In each part, p() = 2 ± c, + c2 [Hint: See the proof of Theorem ] = [a +d) +4bc] nomial, and

3 false, and justify your answer. True-False Exercises a matrix is jo). 9. Prove that the intersection of any two distinct cigenspaces of TF. In parts (a)(fl determine whether the statement is true or bases for the eigenspaces. and the eigenvalues, and then use the method of Example 7 to find TI. For the given matrix, find the characteristic polynomial I Working with Technology = det(p) det() det(p) = det() det(b) = det(p P) = det(p )det()det(p) determinant since erties of the matrix. For example, if we let B = P P, then and B have the same transformations. Such transformations are important because they preserve many prop in which the matrix is mapped into the matrix P P. These are called similarity -* P - P such products, one of which is to view them as transformations Problem will be our main topic of study in this section. There are various ways to think about The Matrix Diagonalization Products of the form pp in which and P are i x n matrices and P is invertible text. significance in a wide \anety of applications, somc of which will be considered later in ibis of and to siinplii various numerical computations. These bases are also of physical of cigenvectors of an?i x n mairix.1. Such bases can be used to study geometric properties In this section we will be concerned with the problem of finding a basis for U that consists 5.2 Diagonalization Theorem for 2 x 2 matrices. is linearly independent. (b) Use the result in Exercise 28 to prove the CayleyHamilton (f) ff is an eigcnvalue of a matrix. then the set of columns of of the reduced row echelon form of. = I (e) The eigenvalues of a matrix are the same as the eigenvalues 1 ing to. (a) Verify the CayleyHamilton Theorem for the matrix corresponding to is the set of eigenvectors of correspond then M +c1 + +Cn. (d) If is an eigenvalue of a matrix, then the eigenspace of + e = p() = 2 + 1, then is invertible. matrix whose characteristic equation is (c) If the characteristic polynomial of a matrix is V. The CayIeHamilton Theorem slates that every square ma (b) If is an eigenvalue of a matrix, then the linear system, then xis an eigenvector of. (a) If is a square matrix and x = x for some nonzero scalar (l )x = has only the trivial solution. trix satisfies its characteristic equation; that is, if is an n x ii I 5 = l IS 2 Chapter 5 Elgenvalues and Eigenvectors V fr

4 2) I 5.2 Dlagonalization 11 Since the eigenvector r+i is nonzero, it follows that Cr+i = But equations (7) and (8) contradict the fact that c1, c2 proof is complete. 4 (8) Cr+i are not all zero so the.1 Exercise Set 5.2 In Exercises [4, show that and B are not similar matrices. 1. =r 11BE1 [ 2J L r4 Il 4 I [2 4j 8=[7 4 = 2 I II [a In Exercises 5H, find a matrix P that diagonalizes, and check your work by computing P P. 5. = 7. [I [6 I, = 9. Let 2 E [ = I = (a) Find the eigenvalues of. (b) For each eigenvalue, find the rank of the matrix l. (c) Is diagonalizable? Justify your conclusion. 1. Follow the directions in Exercise 9 for the matrix 2 12 In Exercises 1114, find the geometric and algebraic multiplic ity of each eigenvalue of the matrix, and determine whether is diagnnalizable. If is diagonalizable, then find a matrix P that diagonalizes, and find P P.. 11.= 1.= = Il 9 [s 14.=I1 5 [o In each part of Exercises I SI 6, the characteristic equation of a matrix is given. Find the size of the matrix and the possible dimensions of its eigenspaces. I 15. (a) ( 1)( + )( 5) = (b) 2( I)( = 16. (a) (1 5 6) = (b) 2 + = In Exercises 17IN, use the method of Example 6 to compute the matrix. I 17. =E r [2 l 19. Let 2. Let =[1 2 l 7 l 1 1 I = 1 and P= I Confirm that P diagonalizes, and then compute ll I = I and P= 1 l 1 Confirm that P diagonalizes, and then compute each of the following powers of. (a) (b) (c) 2 (d) 2 Find un is a positive integer and l = I 2 l I

5 converse is false by showing that the matrices matrices are similar III I I I and B= Oxfl? iusti& your answer. lj [ I] [ I I Ii ol found = is to be p() ( the question and explain answer your reasoning. (a) What can you say about the dimensions of the eigenspaces (b) What can you say about the dimensions of the eigenspaces if you know that is diagonalizable7 (c) If {v1, v,, vi) is a linearly independent set of eigenvectors (a) is diagonalizable if (a (b) is not diagonalizable if(a + 4hc >. of? 111 Section this purpose, assume that is an eigenvalue with geometric eigenspace corresponding to. 29. In thecase where the matrix in Exercise 28 isdiagonalizablc find a matrix P that diagonalizes. LHinr: Sec Exercise uf In Exercises -. find the standard matrix for the given lin. 1. T(xi,x2)=(x,,x1). T(x.x,,.n)= (x1,x,.x.r) * P P (a) Show that S, is a linear operator. (b) Prove that if is diagonalizable. then so is q(). Is the number of nonzero eigenvalues of. 41. TIns problem will lead you through a proof of the fact that q() = c + a,..tfl_t ci + aj matrices Prove that if B = P P, andy is an eigenvector of If diagonalizable, find a matrix P that diagonalizes. integer k. ear operator, and determine whether that matrix is diagonalizablc can be viewed as an operator Sp() = P P on the vector t 22. Show that the matrices (Him: See Exercise 29 of Section 5.1.] of, all of which correspond to the same eigenvalue of, singular matrix? Justify your answer. Justify your answer. and B is similar to C. do you think that must be similar to =F md 8=1 I =P 1 and 8=1 I rank. Show that the converse is false by showing that the 2. We know from Table 1 that similar matrices have the same L J [ same rank but are If then there invertible matrix P P = PB. that there for which Show is no such matrix.] were similar, would be an 2 x 2 have the noi similar. [Suggestion: they 24. We know from Table I that similar matrices have the same eigenvalues. Use the method of Exercise 2 to show that the have the same eigenvalues but are not similar. 25. If. B, and C are n x n matrices such that is similar to B C? Justify your answer. 26. (a) Is it possible for an ii x ii matrix to be similar to itself? (b) What can you say about ann K n matrix that is similar to (c) Is is possible for a nonsingular matrix to be similar to a Suppose that the characteristic 27. polynomial of some matrix 4) I)( J)1( what can you say about that cigenvaluc? 2%. Let that Show 4 4br <. =[(t ] In each part, 12 Chapter 5 Eigenvalues and Elgenvectors multiplicity k. tor of corresponding to. values have the same corresponding eigenvectors for the two the same eigenvalues However, it is not true that those eigen Working with Proofs space Mfl,, of a x a matrices.. T(x1 x) =, (Zr1 x,, 2. T(x:.x2,x) = (8.r + x 4x, 4.tt ± s :) 4. If P is a fixed a x a matrix, then thesimilarity transformation (b) Find the kernel of S (c) Find the rank of S,. 5. Prove that sinular matrices have the same rank and nullity 6. Prove that similar matrices have the same trace Prove that if is diagonalizable, then so is t for every positive 8. We know 1mm Table 1 that similar matrices. and 8, have B corresponding to the eigenvalue. then Pv is the eigenvec 9. Let bean a x a matrix, and let q() be the matrix (a) Prove that if B = P P, then q(b) = Ptq()P. 4. Prove that if is a diagonalizable matrix, then the rank of the algebraic multiplicity of an eigenvalue of anti x a matrix is greater than or equal to the geometric multiplicity. For (a) Prove that therc is a basis B = {u1 U, a,,) for if in which the first k vectors or B form a basis for the v +x1 + it, x1 +.x,) U

6 [flint: Compare the first k column vectors on both sides.l (i) 11 is an eigenvalue ofa matrix, then 2 is singular. L E o YJ then is diagonalizable. X.] (h) If every eigenvalue of a matrix has algebraic multiplicity I. umns. Prove that the product,ip can be expressed as matrix, then is diagonalizable. (b) Let P be the matrix having the vectors in B as col- (g) If there is a basis for R consistingoleigenvectors alan ii x ii 5. ComplexVector Spaces 1 r = a hi is called the complex conjugate oft, = a2 + b2 is called the modulus (or absolute value) oft, respectively, Numbers = a and Imic) = b are called the real part of z and the imaginary part oft, Review of Complex Recall that if z = a + hi is a complex number, then: appears in the hack of this text. study symmetric matrices in more detail. rexiew of the essentials of complex numbers matrices with real entries. In this setion we viii discuss this idea and apph our results to notions of complex eigenv;ilucs and eigenvcciors arise naturally. et en n iihin the context or Because the characteristic equation of any square matrix can hate complex solutions, the 5. Complex Vector Spaces -;Onr (1) lf is diagonalizable, then T is diagonalizable. 2 able. = te) If is diagonalizable and invertible, then 1 is diagonaliz- 1 II 6 (h) nn x a matrix with fewer than a linearly independent eigen (d) If is diagonalizable, then there is a unique matrix P such Ta. Use Theorem to show that the following matrix is not (c) If and B are similar a x a matrices, then there exists an (c) Use the method of Example 6 to cumpute and check your false, and justify your answer, (a) n,z x n matrix with fewer than ii distinct eigenvalues snot diagonalizable. veclors is not diagonalizable. (b) Find a matrix P that diagonalizes. TE. In parts (a)(i) determine whether the statement is true or T2. (a) Use Theorem to show that the following matrix is True-False Exercises ( ) at leastk times, thereby proving that the algebraic (d) By considering det(l C), nomial. (a) determinant. teristic polynomial of C (and hence ) contains the factor invertible it x nmatrix P such that P = BP. result by computing t directly. ( ) ll c nu I and hence that and C have the same characteristic polyt have the same prove that the charac- (b) rank. multiplicity of is greater than or equal Lo the geometric (d) trace. multiplicity k. (e) characteristic polynomial. (f) eigenvalues. diagonalizable. = that PP is diagonal. diagonalizable. L Yj matrix P and then confirm, as stated in Table I, that P P and c = F4 9 TI. Generate a random 4 x 4 matrix and an invertible 4 x 4 Working withtechnology (c) Use the result in part (b) to prove that is similar to