Page 384 - A Course in Linear Algebra with Applications

368 Chapter Nine: Advanced Topics

of the fact that similar matrices have the same minimum poly-
nomial (see 9.4.2).

3. Use 9.4.4 to prove that if an n x n complex matrix has n
distinct eigenvalues, then the matrix is diagonalizable.
4. Show that the minimum polynomial of the companion ma-
trix
/ 0 0 - c
A= 1 0 -b
\ 0 1 -a

is x 3 + ax 2 + bx + c . (See Exercise 8.1.6). [Hint: show that
ul + vA + wA 2 = 0 implies that u — v = w = 0].
5. Find the Jordan normal forms of the following matrices:

b
j
« G 2);< »(-! J)'(«=>(» ~\ j-

6. Read off the minimum polynomials from the Jordan forms
in Exercise 5.
7. Find up to similarity all nxn complex matrices A satisfying
2
A = A .
3
8. The same problem for matrices such that A 2 = A .
9. (Uniqueness of Jordan normal form) Let A be a complex
nxn matrix with Jordan blocks Jij, where J^ is a block
associated with the eigenvalue Cj. Prove that the number of
r x r Jordan blocks Jij for a given i equals d r _i — d r, where
dk is the dimension of the intersection of the column space of
(A — Cil n) k and the null space of A — Cil n. Deduce that the
blocks that appear in the Jordan normal form of A are unique
up to order.
10. Using Exercise 9.4.4 as a model, suggest an n x n matrix
n x
whose minimal polynomial is x n + a n-\x ~ + • • + a,\x + a 0.
•