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# High powers of a square matrix.

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:

Example 11.7.1   The sequences (un) and (vn) are defined by their first terms u0 and v0 and the relations . We wish to know whether these sequences converge , and eventually towards which limits.

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.

Next: The exponential of a Up: Eigenvalues and eigenvectors. Previous: Diagonalization of a square
Noah Dana-Picard
2001-02-26