7.2 Write on paper please Questions: 15, 17, 22, 29 please
Exercise Set 7.2 In Exercises 1-6, find the characteristic equation of the given sym- metric matrix, and then by inspection determine the dimensions of 22. Assuming that b # 0, find a matrix that orthogonally diago- the eigenspaces. nalizes 1 . 18 al 2 . -2 N 23. Let TA : R2 - R' be multiplication by A. Find two orthogo- 1 nal unit vectors u, and u, such that TA (u, ) and TA(u,) are orthogonal. 3. 1 1 [4 1 4 2 2 4 a. A = [ 11 b. A = 2 71 4 2 -1 5. 0 0 -1 2 0 24. Let TA : R3 - R3 be multiplication by A. Find two orthogo- 0 0 0 0 6. 0 2 nal unit vectors u, and u, such that TA(u, ) and TA(uz) are orthogonal. 0 -1 N 2 In Exercises 7-14, find a matrix P that orthogonally diagonalizes A, a. A = 2 4 NI and determine P-1 AP. 12 b. A= 0 1 1 7. A = 6 2V/3 21/3 7 8. A=3 Working with Proofs -2 0 -36 25. Prove that if A is any m x n matrix, then ATA has an ortho- 9. A= - 3 0 10. A = _2 3 normal set of n eigenvectors. - 36 0 -23 26. Prove: If {uj , U2, . . ., u,} is an orthonormal basis for R", and 2 -1 -1 1 if A can be expressed as 11. A = -1 2 -1 12. A = 1 1 0 -1 -1 2 A = cuu + cuu + .. + cu un -7 24 0 0 3 10 07 then A is symmetric and has eigenvalues c,C, . . ., Cn. 24 7 0 0 13. A = 1 30 0 27. Use the result in Exercise 29 of Section 5.1 to prove Theo- 0 0 -7 24 14. A = 0 0 0 0 rem 7.2.2 (a) for 2 X 2 symmetric matrices. 0 0 24 7 10 0 0 0 28. a. Prove that if v is any n X 1 matrix and I is the n x n identity In Exercises 15-18, find the spectral decomposition of the matrix. matrix, then I - vv is orthogonally diagonalizable. b. Find a matrix P that orthogonally diagonalizes I - vv if 15 . 16 . 2 3 - 3 1 -2 -36 v = 17. - 3 ONN 18. - 3 0 N N - 36 0 29. Prove that if A is a symmetric orthogonal matrix, then 1 and -1 are the only possible eigenvalues. In Exercises 19-20, determine whether there exists a 3 X 3 symmet- ric matrix whose eigenvalues are 1 = -1, 12 = 3, 13 = 7 and for 30. Is the converse of Theorem 7.2.2 (b) true? Justify your answer. which the corresponding eigenvectors are as stated. If there is such a 31. In this exercise we will show that a symmetric matrix A is matrix, find it, and if there is none, explain why not. orthogonally diagonalizable, thereby completing the missing 10. *1 = 1- x- [8]-- 181 part of Theorem 7.2.1. We will proceed in two steps: first we will show that A is diagonalizable, and then we will build on that result to show that A is orthogonally diagonalizable. a. Assume that A is a symmetric n x n matrix. One way to prove that A is diagonalizable is to show that for each 20. X1 = eigenvalue do the geometric multiplicity is equal to the algebraic multiplicity. For this purpose, assume that the geometric multiplicity of 1, is k, let Bo = fu , uz, . .., ux} 21. Let A be a diagonalizable matrix with the property that eigen- be an orthonormal basis for the eigenspace correspond- vectors corresponding to distinct eigenvalues are orthogonal. ing to the eigenvalue lo, extend this to an orthonormal Must A be symmetric? Explain your reasoning. basis Bo = {uj, U2, . . ., Un3 for R", and let P be the matrix
