Linear Algebra/Topic: Linear Recurrences/Solutions

SolutionsEdit

Problem 1

Solve each homogeneous linear recurrence relations.

  1.  
  2.  
  3.  
Answer
  1. We express the relation in matrix form.
     
    The characteristic equation of the matrix
     
    has roots of   and  . Any function of the form   satisfies the recurrence.
  2. This is like the prior part, but simpler. The matrix expression of the relation is
     
    and the characteristic equation of the matrix
     
    has the single root  . Any function of the form   satisfies this recurrence.
  3. In matrix form the relation
     
    gives this characteristic equation.
     
Problem 2

Give a formula for the relations of the prior exercise, with these initial conditions.

  1.  ,  
  2.  ,  
  3.  ,  ,  .
Problem 3

Check that the isomorphism given between   and   is a linear map. It is argued above that this map is one-to-one. What is its inverse?

Problem 4

Show that the characteristic equation of the matrix is as stated, that is, is the polynomial associated with the relation. (Hint: expanding down the final column, and using induction will work.)

Problem 5

Given a homogeneous linear recurrence relation  , let  , ...,   be the roots of the associated polynomial.

  1. Prove that each function   satisfies the recurrence (without initial conditions).
  2. Prove that no   is  .
  3. Prove that the set   is linearly independent.
Problem 6

(This refers to the value   given in the computer code below.) Transferring one disk per second, how many years would it take the priests at the Tower of Hanoi to finish the job?