Linear Algebra, Theory And Applications, 2012a

11.4. EXERCISES 285
5. Show that when a Markov matrix is non defective, all of the above theory can be proved
very easily. In particular, prove the theorem about the existence of lim n→∞ A n if the
eigenvalues are either 1 or have absolute value less than 1.
6. Find a formula for A n where
A =
⎛
⎜
⎝
5
2
− 1 2
0 −1
5 0 0 −4
7
2
− 1 2
7
1
2
− 5 2
2
− 1 2
0 −2
Does lim n→∞ A n exist? Note that all the rows sum to 1. Hint: This matrix is similar
to a diagonal matrix. The eigenvalues are 1, −1, 1 2 , 1 2 .
7. Find a formula for A n where
A =
⎛
⎜
⎝
1
2
−1
2 − 1 2
4 0 1 −4
5
2
− 1 2
1 −2
3 − 1 2
1
2
−2
Note that the rows sum to 1 in this matrix also. Hint: This matrix is not similar
to a diagonal matrix but you can find the Jordan form and consider this in order to
obtain a formula for this product. The eigenvalues are 1, −1, 1 2 , 1 2 .
8. Find lim n→∞ A n if it exists for the matrix
⎛
⎞
1
2
− 1 2
− 1 2
0
A = ⎜ − 1 1
2 2
− 1 2
0
⎟
⎝ 1 1 3 ⎠
2
3
2
2
3
2
2
0
3
2
1
The eigenvalues are 1 2
, 1, 1, 1.
9. Give an example of a matrix A which has eigenvalues which are either equal to 1,−1,
or have absolute value strictly less than 1 but which has the property that lim n→∞ A n
does not exist.
10. If A is an n × n matrix such that all the eigenvalues have absolute value less than 1,
show lim n→∞ A n =0.
11. Find an example of a 3 × 3 matrix A such that lim n→∞ A n does not exist but
lim r→∞ A 5r does exist.
12. If A is a Markov matrix and B is similar to A, does it follow that B is also a Markov
matrix?
13. In Theorem 11.1.3 suppose everything is unchanged except that you assume either
∑
j a ij ≤ 1or ∑ i a ij ≤ 1. Would the same conclusion be valid? What if you don’t
insist that each a ij ≥ 0? Would the conclusion hold in this case?
14. Let V be an n dimensional vector space and let x ∈ V and x ≠ 0. Consider β x ≡
x,Ax, ··· ,A m−1 x where
A m x ∈ span ( x,Ax, ··· ,A m−1 x )
⎞
⎟
⎠
⎞
⎟
⎠