Math 3353, Spring 2015
Due April 8
Homework 10 – Diagonalization and Complex Eigenvalues
1.
Determine whether the following matrices are diagonalizable, are not diagonalizable, or
that you have insufficient information to determine; justify each answer.
(a) A ∈ R5×5 has two distinct eigenvalues. One eigenspace is two-dimensional, the other is
three-dimensional.
(b) A ∈ R3×3 has two distinct eigenvalues. Both eigenspaces are one-dimensional.
(c) A ∈ R4×4 has three distinct eigenvalues. One eigenspace is two-dimensional.
(d) A ∈ R7×7 has three distinct eigenvalues. One eigenspace is three-dimensional.
(e) A ∈ R4×4 has four distinct eigenvalues.
2.

1
0
Diagonalize the matrix A = 
0
1
0
2
0
0
0
0
2
0

0
0
.
0
3
3.
complex) eigenvalues and a basis for each eigenspace for the matrix
 Find the (possibly

4 0
0
A =  0 1 −2 
0 1
3
4.
Note: for this problem we will use the book’s notation, in which a is the complex conjugate
of a ∈ C and x is the complex conjugate of x ∈ Cn .
A matrix A ∈ Rn×n is symmetric if A = AT . For any complex vector x ∈ Cn we may define the
scalar q = xT Ax. The proof below guarantees that q is a real number by showing that q = q.
Justify each step of this proof.
q = xT Ax
(a) justification
= xT Ax
(b) justification
= xT Ax
(c) justification
= xT Ax
= xT Ax
T
(d) justification
T
T
T T
(e) justification
= x A (x )
(f) justification
= x T AT x
(g) justification
T
= x Ax
(h) justification
=q
(i) justification