The nth Power of a Matrix and Approximations for Large n


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New Zealand Journal of Mathematics

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

C.S. Withers

Applied Mathematics Group

Industrial Research Limited

Lower Hutt


S. Nadarajah

School of Mathematics

University of Manchester

Manchester M13 9PL


Abstract When a square matrix A, is diagonalizable, (for example, when A is Hermitian or has distinct eigenvalues), then An can be written as a sum of the nth powers of its eigenvalues with matrix weights. However, if a 1 occurs in its Jordan form, then the form is more complicated: An 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, An is proportional to nm − 1λn 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.

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