2006 SMT/Advanced Topics Problems/Problem 2
Problem
Define . Find a vertical vector such that (where is the identity matrix).
Solution
If we calculate the first couple powers of , we quickly see a pattern:
Lemma:
Proof: Clearly this is true for . Thus, we proceed with an induction arguement.
Thus, our proof is complete.
Therefore, and this is equal to .
Finally, we want to find a vector such that . Letting , we get the system of equations and . From this, we have and , and so our desired vector is .