Linear Algebra. Math-4100, Fall 2007
Assignment 8
Due Monday, November 19, by 4pm. (Either in class, or my mailbox in AE 301, or
under my door AE 405).
Reading
Nov. 5, 8, 12, and 15: Strang Sections 6.2–6.4, 6.6, 10.2, 6.5.
Problems
You are encouraged to consult the text and notes and discuss the problems with other people.
However, the solutions should be yours. Please indicated on your papers, who you discussed
the problems with.
1. Find Ato change the scalar equation y00
−5y0+ 4y= 0 into a vector equation for
u= [y, y0]T,
d
dtu=Au
What are the eigenvalues of A? Find them also by substituting y=eλt into the scalar
equation.
2. Problems 6.3 #19–20
3. Problem 6.3 #17, 21.
4. Problem 6.4 #4.
5. Problem 6.4 #15.
6. Problem 6.4 #20.
7. Let A=2a
2 0 .What number amakes A=QΛQTpossible? What number makes
A=SΛS−1impossible? What number makes A−1impossible?
8. Problem 6.6 #3. A fool-proof way to show the similarity of Aand Bis to diagonal-
ize them A=F−1ΛF, B =G−1ΛG; then Λ = F−1AF, and B=G−1F−1AF G =
(F G)−1A(F G).For these examples there is an easier way. No matter which method
you use, please check that B=M−1AM for Myou found.
9. Problem 6.6 #20.
10. Problem 10.2 #30.
11. (10.2 #28). Use the definition to demonstrate that if A+iB is a Hermitian matrix
(Aand Bare real) then AT=Aand BT=−B(a sign typo in my original text is
corrected here). Therefore A−B
B A is symmetric.
