Math 102  Homework 8 (selected problems)
David Lipshutz
Problem 1.
(Strang, 5.5: #14)
In the list below, which classes of matrices contain
A
and which contain
B
?
A
=
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
and
B
=
1
4
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
Orthogonal
,
invertible
,
projection
,
permutation
,
Hermitian
,
rank
1,
diagonalizable
,
Markov
.
Find the eigenvalues of
A
and
B
.
Proof.
A
is orthogonal, invertible, not projection (since
A
6
=
A
2
), permutation, not Hermitian
(since
A
6
=
A
T
), diagonalizable and Markov. The characteristic polynomial for
A
is
p
(
λ
) =
λ
4

1, so the eigenvalues of
A
are
±
1 and
±
i
.
B
is projection (onto the space spanned by
(1
,
1
,
1
,
1)), Hermitian, rank1, diagonalizable, Markov. The eigenvalues of
B
are 1 since it is
a Markov matrix and the rest 0 since it is rank1.
Problem 2.
(Strang, 5.5: #16)
Write one signiﬁcant fact about the eigenvalues of each of the following.
(a) A real symmetric matrix.
(b) A stable matrix: all solutions to
du/dt
=
Au
approach zero.
(c) An orthogonal matrix.
(d) A Markov matrix.
(e) A defective matrix (nondiagonalizable).
(f) A singular matrix.
Proof.
(a) Every eigenvalue is real.
(b) All eigenvalues have norm less than 1.
(c) Eigenvalues all have norm equal to 1.
(d) 1 is an eigenvalue of the matrix and the other eigenvalues are less than 1.
(e) The matrix has repeated eigenvalues.
(f) 0 is an eigenvalue of the matrix.
1