Suppose the migration matrix for three locations is
( )
.5 0 .3
|( .3 .8 0 |) .
.2 .2 .7
Find a comparison for the populations in the three locations after a long time.
Show that if ∑iaij = 1, then if A =
(aij)
, 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 so.
If A satisfies the conditions of the above problem, can it be concluded that limn→∞An
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 limn→∞An if the eigenvalues are either 1 or
have absolute value less than 1.
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,
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
exist.
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 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 12.1.3 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
m− 1
βx ≡ x, Ax, ⋅⋅⋅,A x
where
m ( m−1 )
A x ∈ span x,Ax,⋅⋅⋅,A x
and m is the smallest such that the above inclusion in the span takes place. Show that
{x,Ax, ⋅⋅⋅,Am− 1x }
must be linearly independent. Next suppose
{v ,⋅⋅⋅,v }
1 n
is a basis for V .
Consider βvi as just discussed, having length mi. Thus Amivi is a linearly combination
of vi,Avi,
⋅⋅⋅
,Am−1vi for m as small as possible. Let pvi
(λ)
be the monic polynomial
which expresses this linear combination. Thus pvi
(A)
vi = 0 and the degree of pvi
(λ)
is
as small as possible for this to take place. Show that the minimum polynomial for A
must be the monic polynomial which is the least common multiple of these polynomials
pvi