Linear Algebra and Analysis

12.6 Exercises

1. 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.

2. Show that if ∑ _ia_ij = 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 a_ij ≥ 0 for this to be so.
3. If A satisfies the conditions of the above problem, can it be concluded that lim_n→∞Aⁿ exists?
4. Give an example of a non regular Markov matrix which has an eigenvalue equal to −1.
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→∞Aⁿ if the eigenvalues are either 1 or have absolute value less than 1.
6. Find a formula for Aⁿ where

   ( 5 1 ) 2 − 2 0 − 1 || 5 0 0 − 4 || A = |( 72 − 12 12 − 52 |) 7 − 1 0 − 2 2 2

   Does lim_n→∞Aⁿ 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 Aⁿ where

   ( ) 2 − 12 12 − 1 || 4 0 1 − 4 || A = |( 5 − 1 1 − 2 |) 2 21 1 3 − 2 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→∞Aⁿ if it exists for the matrix

   ( 1 1 1 ) 2 − 2 − 2 0 A = || − 12 12 − 12 0 || |( 12 12 32 0 |) 3 3 3 1 2 2 2

   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→∞Aⁿ 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→∞Aⁿ = 0.
11. Find an example of a 3 × 3 matrix A such that lim_n→∞Aⁿ 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 12.1.3 suppose everything is unchanged except that you assume either ∑ _ja_ij ≤ 1 or ∑ _ia_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

    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 β_vi as just discussed, having length m_i. Thus A^m_iv_i is a linearly combination of v_i,Av_i,
    ⋅⋅⋅
    ,A^m−1v_i for m as small as possible. Let p_vi
    (λ)
    be the monic polynomial which expresses this linear combination. Thus p_vi
    (A)
    v_i = 0 and the degree of p_vi
    (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 p_vi
    (λ)
    .