Let . By induction, we prove that .
Let A be a square matrix; we wish to compute An for some positive integer n. If n is a large number, this is a tedious task. Suppose now that A is diagonalizable, i.e. there exists an invertible matrix P such that D=P-1AP is a diagonal matrix. Then A= PDP-1 and we have:
Write the given relations in matricial form: .
By induction, we show that .
The characteristic polynomial of A is . the matrix A has two eigenvalues, namely and , therefore A is diagonalizable and is similar to . We denote by P the change of basis matrix.
Then we have: An= PDnP-1 and .
As , we have: , therefore . As a consequence, we have that the sequences (un) and (vn) are convergent and have 0 as their common limit.