Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
If C = F-1 AF and also C = G-1 BG, what matrix M gives B = M-t AM? Conclusion: If C is similar to A and also to B then

Introduction to Linear Algebra 4
If A = diag(1, 3) and B = diag(3, 1) show that A and B are similar (find an M

Introduction to Linear Algebra 4
Show that A and B are similar by finding M so that B = M- 1 AM: A = [! ~] A = [! !] A = [~ ~] and and and B = [~ ~] B = [ 1 -1] -1 1 B = [~ ~l

Introduction to Linear Algebra 4
If a 2 by 2 matrix A has eigenvalues 0 and 1, why is it similar to A = [A 8]? Deduce from Problem 1 that all 2 by 2 matrices with those eigenvalues are similar.

Introduction to Linear Algebra 4
Which of these six matrices are similar? Check their eigenvalues. [~~] [~~] [~~] [~~] [! ~] [~ !].

Introduction to Linear Algebra 4
There are sixteen 2 by 2 matrices whose entries are O's and 1 'so Similar matrices go into the same family. How many families? How many matrices (total 16) in each family?

Introduction to Linear Algebra 4
(a) If x is in the nullspace of A show that M- 1 x is in the nullspace of M- 1 AM. (b) The nullspaces of A and M-l AM have the same (vectors)(basis)(dimension).

Introduction to Linear Algebra 4
Suppose Ax = Ax and Bx = Ax with the same A's and x's. With n independent eigenvectors we have A = B: Why? Find A '# B when both have eigenvalues 0,0 but only one line of eigenvectors (Xl, 0).

Introduction to Linear Algebra 4
By direct multiplication find A 2 and A 3 and AS when A=[~ ~J. Guess the form of A k. Set k = 0 to find A 0 and k = -1 to find A-I

Introduction to Linear Algebra 4
Questions 10-14 are about the Jordan form.

Introduction to Linear Algebra 4
Questions 10-14 are about the Jordan form.

Introduction to Linear Algebra 4
Questions 10-14 are about the Jordan form.

Introduction to Linear Algebra 4
Questions 10-14 are about the Jordan form.

Introduction to Linear Algebra 4
Questions 10-14 are about the Jordan form.

Introduction to Linear Algebra 4
Prove that det(A - AI) = det(M- 1 AM - AI). (You could write I = M- 1 M and factor out det M- 1 and det M.) Since these characteristic polynomials of A and M- 1 AM are the same, the eigenvalues are the same (with the same multiplicities).

Introduction to Linear Algebra 4
Which pairs are similar? Choose a, b, c, d to prove that the other pairs aren't:

Introduction to Linear Algebra 4
True or false, with a good reason: (a) A symmetric matrix can't be similar to a nonsymmetric matrix. (b) An invertible matrix can't be similar to a singular matrix. (c) A can't be similar to -A unless A = 0. (d) A can't be similar to A + I.

Introduction to Linear Algebra 4
If B is invertible, prove that AB is similar to BA. They have the same eigenvalues.

Introduction to Linear Algebra 4
If A is 6 by 4 and B is 4 by 6, A Band BA have different sizes. But with blocks M-1FM = [~ -1J [A! 6J [~ 1J = [~ B~J = G. (a) What sizes are the four blocks (the same four sizes in each matrix)? (b) This equation is M- 1 FM = G, so F and G have the same 10 eigenvalues. F has the 6 eigenvalues of AB p

Introduction to Linear Algebra 4
Why are these statements all true? (a) If A is similar to B then A 2 is similar to B2. (b) A2 and B2 can be similar when A and B are not similar (try A = 0,0). (c) [i ~] is similar to [i !]. (d) [i ~] is not similar to [i ~ ]. (e) If we exchange rows 1 and 2 of A, and then exchange columns 1 and 2, t

Introduction to Linear Algebra 4
If J is the 5 by 5 Jordan block with A = 0, find J2 and count its eigenvectors and find its Jordan form (there will be two blocks)

Introduction to Linear Algebra 4
If an n by n matrix A has all eigenvalues A = 0, prove that An = zero matrix. (Maybe prove first that J n = zero matrix, by direct multiplication. Or use the CayleyHamilton Theorem?)

Introduction to Linear Algebra 4
For the shifted QR method in the Worked Example 6.6 B, show that A2 is similar to A 1. No change in eigenvalues, and the A's quickly approach a diagonal matrix.

Introduction to Linear Algebra 4
If A is similar to A-I, must all the eigenvalues equal 1 or -1 ?