Math 20820 Statistics Worksheet
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2Math 20820: Homework 2
Due Wednesday, Jan. 27.
1. Let M
Mat(m, n, R) be a Markov (or stochastic) matrix. That is all the entries
of M are in the interval [0, 1]
R and the sum of each column adds to one.
(a) Let
R be a Markov (or stochastic) vector. That is
= (v
, v
, . . . , v )
1
2
with v
[0, 1] for each j = 1, . . . , n and v
+ v
+
+ v = 1. Prove that M is a
1
2
stochastic vector in R .
(b) Prove that if M, N
Mat(n, n, R) are stochastic then M N is stochastic.
(c) Prove that M is stochastic then as well.
(d) Is the subset of stochastic matrices in Mat(n, n, R) a subspace of Mat(n, n, R)?
Why or why not.
2. Last semester, in Homework 11, you did problem #9 in Section 5.A of Axler,
3
which had you find all the eigenvalues and eigenvectors for T
(F
) given by
T (z
, z
, z
) = (2z
, 0, 5z
) for F = R or C. Let
= (1, 0, 0),
= (0, 1, 0), and
1
2
3
2
3
1
2
3
= (0, 0, 1) and denote by
= (
,
,
) the standard basis for F
.
3
1
2
3
(a) What is [T ] ?
(b) Is [T ] diagonalizable? Why or why not?
2
(c) Show that T
(T
5I) = 0 but that T (T
5I) = 0.
3. Let F = Z for p a prime. For what p do your answers to #2 above change for
3
T
(F
) given by T (z
, z
, z
) = (2z
, 0, 5z
)? How are your answers different?
1
2
3
2
3
3
3
4. Let T
(F
) such that in the standard basis
= (
,
,
) for F
1
2
3
1 1 1
[T ] =
0 1 1
0 0 1
(a) Give all: (i) eigenvalues of T ; (ii) all eigenvectors of T ; and (iii) all generalized
eigenvectors of T .
(b) Is T diagonalizable? Why or why not?
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