# The nth Power of a Matrix and Approximations for Large n

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 38, (2008), **Pages** 171-178

**C.S. Withers**

Applied Mathematics Group

Industrial Research Limited

Lower Hutt

NEW ZEALAND

**S. Nadarajah**

School of Mathematics

University of Manchester

Manchester M13 9PL

UNITED KINGDOM

mailto:mbbsssn2@manchester.ac.uk

**Abstract** When a square matrix *A*, is diagonalizable, (for example,
when *A* is Hermitian or has distinct eigenvalues), then *A*^{n}
can be written as a sum of the *n*th powers of its eigenvalues with matrix weights. However, if a 1 occurs in its Jordan form, then the form is more complicated: *A*^{n} can be written as a sum of polynomials of degree *n* in its eigenvalues with coefficients depending on *n*.
In this case to a first approximation for large *n*, *A*^{n}
is proportional to *n*^{m − 1}λ^{n} with a constant matrix multiplier, where λ is the eigenvalue of maximum modulus and *m* is the maximum multiplicity of λ.

**Keywords** powers of matrices, approximations, Jordan form, Singular value decomposition.

**Classification** (MSC2000) Primary 15A21 Canonical forms, reductions, classification
Secondary 15A24 Matrix equations and identities.