Linear Algebra 2

Department of Mathematics, University of Toronto MAT224H1F - Linear Algebra II

Fall 2013

Problem Set 5

• Due at 12:00 noon on Wednesday, December 4, in the drop boxes in SS1071.

• Be sure to clearly write your name, student number, your tutorial group and the name of your TA on the top right-hand corner of your assignment, and staple the pages together. Remember that the clarity of your solutions matters - your assignment should be a final draft, not a first draft.

• You are welcome to work in groups but problem sets must be written up independently - any suspicion of copying/plagiarism will be dealt with accordingly. You are welcome to discuss the problem set questions in tutorial, or with your instructor. You may also use Piazza to discuss problem sets but you are not permitted to ask for or post complete solutions to problem set questions.

1. Consider the matrix

A =

 

1 0 2 5 0 0 i 1 0 0 0 −1 0 0 0 −1

  .

Compute A100. Justify your answer.

2. Let N : C4 → C4 be the linear function whose matrix with respect to the standard basis is 

1 + i −1 − i i 0 i −i −1 + i 0 −1 1 −1 0 −1 1 −1 0

  .

(a) Show that N is nilpotent, and find the smallest k such that Nk = 0.

(b) Find a canonical basis α, and determine [N]αα.

3. Let T : C4 → C4 be the linear function whose matrix with respect to the standard basis is  −1 −1 5 6 0 −1 4 8 0 0 3 0 0 0 0 3

  .

Find a basis α such that [T ]αα is the Jordan canonical form of T, and find [T ] α α.

4. Let T : C4 → C4 be the linear function whose matrix with respect to the standard basis is 

3 1 5 −4 0 3 −3 3 0 0 9 −6 0 0 9 −6

  .

Find a basis α of C4 such that [T ]αα is the Jordan canonical form of T , and find [T ] α α.

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5. Let V be a finite-dimensional vector space over C, with inner product 〈·, ·〉. Let T : V → V be a linear function, and let W be a subspace of V . Prove that W is T-invariant if and only if W⊥ is T∗-invariant.

6. Recall that the trace of an n×n matrix is the sum of the diagonal entries of the matrix. In this problem you may use, without proof, the fact that for any n×n matrices A and B, trace(AB) = trace(BA).

(a) Let U ∈ Mn×n(C) be an upper triangular matrix, and let λ1, . . . ,λn be the (not necessarily distinct) eigenvalues of U. Prove that trace(U) = λ1 + · · · + λn.

(b) Suppose that A,B ∈ Mn×n(C) are similar. Prove that trace(A) = trace(B). (c) Let A be an n×n matrix over C, and let λ1, . . . ,λn be the (not necessarily distinct) eigenvalues

of A. Prove that trace(A) = λ1 + · · · + λn.

7. (a) Let A be an n×n rank 1 matrix in Jordan form. Prove that (up to reordering the Jordan blocks) A has one of the following two forms:

 λ 0 · · · 0 0 0 · · · 0 ...

... · · · ...

0 0 · · · 0

  , for some λ 6= 0 .

 

0 1 0 · · · 0 0 0 0 · · · 0 ...

... ... · · ·

... 0 0 0 · · · 0

  .

(b) Suppose that V is an n-dimensional vector space over C, and that T : V → V is linear. Prove that if dim ker(T) = n − 1, then T is either nilpotent or diagonalizable. Hint: Consider [T]αα, where α is a canonical basis.

8. (a) Let B be an n×n Jordan block with 0 on the diagonal. Let α = {e1, . . . ,en} be the standard basis, and let T : Cn → Cn be the linear function such that [T ]αα = B. Let β = {en, . . . ,e1 } be the basis α, written in reverse order. Prove that [T ]

β β = B

t.

(b) Let J be an n×n Jordan block. Prove that J is similar to Jt. Hint: Write J = λI + B, where λ is the value on the diagonal of J, and B is an n × n Jordan block with 0 on the diagonal.

(c) Let J be an n×n matrix in Jordan form. Prove that J is similar to Jt. Hint: You may, without proof, use the last problem from the previous problem set applied to k matrices instead of 2. You may also use, without proof, the fact that for any matrices A1,B1 ∈ Ml1×l1 (C), . . . ,Ak,Bk ∈ Mlk×lk (C),

(A1 ⊕···⊕Ak)(B1 ⊕···⊕Bk) = (A1B1) ⊕···⊕ (AkBk).

(d) Prove that if A ∈ Mn×n(C) then A is similar to At. Hint: We saw in class that A is similar to its Jordan form.

Suggested Extra Problems (not to be handed in):

• Textbook, Section 6.1 1, 3, 5, 8, 10, 11

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• Textbook, Section 6.2 1, 2, 7, 12, 13

• Textbook, Section 6.3 1, 2, 4, 5, 7

• Textbook, Section 6.4 1, 4, 5

• Textbook, Section 6.5 3, 5, 9

• Textbook, Chapter 6 Supplementary Exercises 4, 5, 6, 10, 11

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