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
.