# [R] Multiplication of Matrices

Prof Brian D Ripley ripley at stats.ox.ac.uk
Wed Oct 9 13:27:27 CEST 2002

```On Wed, 9 Oct 2002, Ko-Kang Kevin Wang wrote:

> Suppose I have a matrix, A.  Is there an easy way to find A^{n}?
>
> I mean, I can do something like:
>   A %*% A %*% A %*% A
> for A^4, but if I want A^{10} it would be kind of annoying...

Use the eigendecomposition.  If A = V %*% diag(lam) %*% t(V), then
A^n = V %*% diag(lam^n) %*% t(V).  More care (including complex conjugates)
is needed if the eigendecomposition is complex, but the method still works.

--
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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```