Linear Algebra. Math-4100, Fall 2007
Assignment 7
Due Thursday, November 8, by 4pm. (Either in class, or my mailbox in AE 301, or
under my door AE 405).
Reading
Oct. 29, Nov. 1: Strang Sections 10.1-10.3, 6.1.
Nov. 5 and Nov. 8: Strang Sections 6.2–6.5.
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. Please submit extra credit problems on a separate sheet of paper.
1. Problems 10.2 #3, 12, and 22.
2. Problems 6.1 #2, 5
3. Problem 6.1 #14
4. Problem 6.1 #22.
5. Problem 6.1 #28.
6. Consider the matrix
A=
0a a ... a
a0a ... a
a a 0... a
. . . ... .
a a a ... 0
(a) Show that [1,1, ..., 1]Tis an eigenvector; find the corresponding eigenvalue, λ1.
(b) Show that λ2=−ais an eigenvalue; find the corresponding eigenvectors; there
are many.
7. The Caley-Hamilton Theorem: Consider the characteristic polynomial of A, p(x) =
det(A−xI).For a n×nmatrix, p(x) is a polynomial of degree n, p(x) = c0+c1x+
... +cnxn.The Caley-Hamilton Theorem says that c0I+c1A+... +cnAn= 0,i.e.,
p(A) = 0. Read Problem 6.2.35 that gives a proof of the theorem for a diagonalizable
matrix. Don’t hand in anything for this problem.
8. Problem 6.2.36.
9. Consider the matrix (a Jordan block) with a’s on the principal diagonal and 1’s on the
next diagonal; all other entries are zeros:
A=
a1
a1
a1
a1
a
