Page 283 - 35Linear Algebra

15.1 Review Problems 283

(b) Explain why there exist scalars α i not all zero such that

2
n
α 0 v + α 1 Lv + α 2 L v + · · · + α n L v = 0 .
(c) Let m be the largest integer such that α m 6= 0 and

2
n
p(z) = α 0 + α 1 z + α 2 z + · · · + α m z .
Explain why the polynomial p(z) can be written as

p(z) = α m (z − λ 1 )(z − λ 2 ) . . . (z − λ m ) .

[Note that some of the roots λ i could be complex.]

(d) Why does the following equation hold

(L − λ 1 )(L − λ 2 ) . . . (L − λ m )v = 0 ?

(e) Explain why one of the numbers λ i (1 ≤ i ≤ m) must be an
eigenvalue of L.

4. (Dimensions of Eigenspaces)

(a) Let
 
4 0 0
A =  0 2 −2  .
0 −2 2
Find all eigenvalues of A.

(b) Find a basis for each eigenspace of A. What is the sum of the
dimensions of the eigenspaces of A?
(c) Based on your answer to the previous part, guess a formula for the
sum of the dimensions of the eigenspaces of a real n×n symmetric
matrix. Explain why your formula must work for any real n × n
symmetric matrix.

T
5. If M is not square then it can not be symmetric. However, MM and
T
M M are symmetric, and therefore diagonalizable.
(a) Is it the case that all of the eigenvalues of MM T must also be
T
eigenvalues of M M?

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