- Suppose the migration matrix for three locations is
Find a comparison for the populations in the three locations after a long time.

- Show that if ∑
_{i}a_{ij}= 1, then if A =, then the sum of the entries of Av equals the sum of the entries of v. Thus it does not matter whether a_{ij}≥ 0 for this to be so. - If A satisfies the conditions of the above problem, can it be concluded that lim
_{n→∞}A^{n}exists? - Give an example of a non regular Markov matrix which has an eigenvalue equal to −1.
- 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. - Find a formula for A
^{n}whereDoes 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,,. - Find a formula for A
^{n}whereNote 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,

,. - Find lim
_{n→∞}A^{n}if it exists for the matrixThe eigenvalues are

,1,1,1. - 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. - If A is an n × n matrix such that all the eigenvalues have absolute value less than 1, show
lim
_{n→∞}A^{n}= 0. - 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. - If A is a Markov matrix and B is similar to A, does it follow that B is also a Markov matrix?
- In Theorem 10.1.3 suppose everything is unchanged except that you assume either
∑
_{j}a_{ij}≤ 1 or ∑_{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? - Let V be an n dimensional vector space and let x ∈ V and x≠0. Consider
β
_{x}≡ x, Ax,,A^{m−1}x whereand m is the smallest such that the above inclusion in the span takes place. Show that

must be linearly independent. Next supposeis a basis for V . Consider β_{vi}as just discussed, having length m_{i}. Thus A^{mi}v_{i}is a linearly combination of v_{i},Av_{i},,A^{m−1}v_{i}for m as small as possible. Let p_{vi}be the monic polynomial which expresses this linear combination. Thus p_{vi}v_{i}= 0 and the degree of p_{vi}is as small as possible for this to take place. Show that the minimal polynomial for A must be the monic polynomial which is the least common multiple of these polynomials p_{vi}. - If A is a complex Hermitian n×n matrix which has all eigenvalues nonnegative, show that there exists a complex Hermitian matrix B such that BB = A.
- ↑Suppose A,B are n × n real Hermitian matrices and they both have all nonnegative
eigenvalues. Show that det≥ det+ det. Hint: Use the above problem and the Cauchy Binet theorem. Let P
^{2}= A,Q^{2}= B where P,Q are Hermitian and nonnegative. Then - Suppose B = is an×Hermitian nonnegative matrix where α is a scalar and A is n×n. Show that α must be real, c = b, and A = A
^{∗},A is nonnegative, and that if α = 0, then b = 0. Otherwise, α > 0. - ↑If A is an n × n complex Hermitian and nonnegative matrix, show that there exists an
upper triangular matrix B such that B
^{∗}B = A. Hint: Prove this by induction. It is obviously true if n = 1. Now if you have an×Hermitian nonnegative matrix, then from the above problem, it is of the form,α real. - ↑ Suppose A is a nonnegative Hermitian matrix (all eigenvalues are nonnegative) which is
partitioned as
where A

_{11},A_{22}are square matrices. Show that det≤ detdet. Hint: Use the above problem to factor A gettingNext argue that A

_{11}= B_{11}^{∗}B_{11},A_{22}= B_{12}^{∗}B_{12}+ B_{22}^{∗}B_{22}. Use the Cauchy Binet theorem to argue that det= det≥ det. Then explain why - ↑ Prove the inequality of Hadamard. If A is a Hermitian matrix which is nonnegative (all
eigenvalues are nonnegative), then det≤∏
_{i}A_{ii}.

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