Math 20820: Homework 1 - University Of Notre Dame Page 3
ADVERTISEMENT
1
2
33
(k) Note that T is stochastic and identify the stable equilibrium distribution.
10
11
(l) What is T
? What is T
? What is T as n
?
3. If we change the field in problem #2 above to Z
does it change the results?
2
2
How? That is let T
(Z
) be given by T (w, z) = (z, w). Is there a basis in which
2
T is diagonalizable? Why or why not?
2
4. Consider the stochastic matrix in the standard basis for R
:
1/2 1/4
A =
.
1/2 3/4
(a) Find the eigenvalues and eigenspaces of A and identify the stable equilibrium
distribution.
(b) Verify that A is diagonalizable and find a basis of eigenvectors. Compute the
change of basis matrices
and
.
(c) Compute the trace of A and the sum of the eigenvalues of A.
(d) Compute the determinant of A and the product of the eigenvalues of A.
10
(e) What is A
? What is A as n
?
5. Let T
(V ).
2
(a) How are the eigenvalues of T
related to the eigenvalues of T ?
2
(b) How are the eigenvectors of T
related to the eigenvectors of T ?
(c) What about the eigenvalues and eigenvectors of T compared to T ?
6. Consider
a b
A =
Mat(2, 2, C),
c d
and suppose that λ
and λ
are the (not necessarily distinct) eigenvalues of A.
1
2
(a) Prove that the trace of A, i.e. a + d, is equal to λ
+ λ
.
1
2
(b) Prove that the determinant of A, i.e. ad
bc is equal to λ
λ
.
1
2
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education