Linear Algebra Notes Chapter 9 MULTIPLE EIGENVALUES AND ...

More documents

Recommendations

Info

6 as claimed. □ A matrix is called nilpotent if some power of it is zero. It turns out, for 2 × 2 matrices Y as we are considering here, that if Y n = 0, then already Y 2 = 0 (see exercise 9.4). So a 2×2 matrix is nilpotent if either it is already zero, or if it squares to zero. We may summarize the main points in this chapter as follows. Any matrix A with a multiple eigenvalue λ is a scalar matrix λI plus a nilpotent matrix A 0 = A − λI. If [ A is] not itself a scalar matrix, [ then] A 0 ≠ 0. In that case A is conjugate to λ 1 0 1 , and A 0 λ 0 is conjugate to . This last matrix does not involve λ. In 0 0 particular, all matrices A with multiple [ eigenvalues ] have their nilpotent parts A 0 0 1 being conjugate to the same matrix . 0 0 Exercise 9.1. For each of the[ following ] matrices A, find the eigenvalue λ, and a λ 1 matrix B such that B −1 AB = . 0 λ [ ] 6 1 (a) A = [ −1 8 ] 1 −1 (b) A = [ 1 −1 ] 1 0 (c) A = [ 1 1 ] 0 1 (d) A = −1 2 (To make life easier for the grader, please choose v = e 1 in each problem. Your B’s will all be the same, having 1 in the upper right corner.) [ ] λ 1 Exercise 9.2. Show that every eigenvector of is proportional to e 0 λ 1 . (Hint: Just calculate.) Exercise 9.3. Let A be a matrix with multiple eigenvalue λ, and let u be an eigenvector of A. Assume that A is not a scalar matrix. Prove the following statements. (a) Every eigenvector of A is proportional to u. (Hint: Proposition 1.) (b) A 0 u = (0, 0) (c) If v is a vector for which A 0 v = (0, 0), then v is an eigenvector of A. (Taken together, these facts mean that A has exactly one eigenline, which consists precisely of the vectors sent to (0, 0) by A 0 . ) [ ] 0 0 Exercise 9.4. Suppose A n = for some n ≥ 0. 0 0 (a) What are the eigenvalues of A? (Hint: Apply A repeatedly to both sides of the equation Au = λu. ) (b) Compute A 2 . (Hint: Use Proposition 3, and the value of λ from part (a). ) Exercise 9.5. Suppose λ = 0 is the only eigenvalue of A. Does this imply that A is nilpotent?
7 Exercise 9.6. Show that the nilpotent matrices are precisely those of the form [ ] a b A = c −a with a 2 + bc = 0. (You have to show two things: First, that every nilpotent matrix has this form. Second, that every matrix of this form is nilpotent.)
Page 1 and 2: Linear Algebra Notes Chapter 9 MULT
Page 3 and 4: 3 as claimed in Proposition 2. □
Page 5: 5 But also A = λI + A 0 , so λI +

Linear Algebra Notes Chapter 9 MULTIPLE EIGENVALUES AND ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?