- 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 ∑
iaij = 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 aij ≥ 0 for this to be
- If A satisfies the conditions of the above problem, can it be concluded that limn→∞An
- Give an example of a non regular Markov matrix which has an eigenvalue equal to
- 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 limn→∞An if the
eigenvalues are either 1 or have absolute value less than 1.
- Find a formula for An where
Does limn→∞An 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 An where
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,
- Find limn→∞An if it exists for the matrix
The eigenvalues are
- 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 limn→∞An does not
- If A is an n × n matrix such that all the eigenvalues have absolute value less than 1, show
limn→∞An = 0.
- Find an example of a 3 × 3 matrix A such that limn→∞An does not exist but limr→∞A5r
- If A is a Markov matrix and B is similar to A, does it follow that B is also a Markov
- In Theorem 10.1.3 suppose everything is unchanged except that you assume either
jaij ≤ 1 or ∑
iaij ≤ 1. Would the same conclusion be valid? What if you don’t insist that
each aij ≥ 0? Would the conclusion hold in this case?
- Let V be an n dimensional vector space and let x ∈ V and x≠0. Consider
βx ≡ x, Ax,
and m is the smallest such that the above inclusion in the span takes place. Show that
must be linearly independent. Next suppose
is a basis for
V . Consider βvi as just discussed, having length mi. Thus Amivi is a linearly combination of
,Am−1vi for m as small as possible. Let pvi be the monic polynomial which
expresses this linear combination. Thus
vi = 0 and the degree of pvi 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
- 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 P2 = A,Q2 = B where P,Q are Hermitian and nonnegative.
- Suppose B = is an
× Hermitian nonnegative matrix where
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
nonnegative matrix, then from the above problem, it is of the form
- ↑ Suppose A is a nonnegative Hermitian matrix (all eigenvalues are nonnegative) which is
where A11,A22 are square matrices. Show that det
Hint: Use the
above problem to factor A getting
Next argue that A11 = B11∗B11,A22 = B12∗B12 + B22∗B22. Use the Cauchy Binet theorem
to argue that 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