Let . By induction, we prove that .

Let *A* be a square matrix; we wish to compute *A*^{n} 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*^{-1}*AP* 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:
*A*^{n}= *PD*^{n}*P*^{-1} and
.

As
,
we have:
,
therefore
.
As a
consequence, we have that the sequences (*u*_{n}) and (*v*_{n}) are convergent and have 0 as their common limit.