The nth Power of a Matrix and Approximations for Large n - NZJM

New Zealand Journal of Mathematics

C.S. Withers¹  S. Nadarajah² 

[1] Applied Mathematics GroupIndustrial Research Limited Lower Hutt NEW ZEALAND$$;School of MathematicsUniversity of ManchesterManchester M13 9PLUNITED KINGDOM$$

关键词: powers of matrices;    approximations;    Jordan form;    Singular value decomposition.;   

DOI  :  

学科分类：社会科学、人文和艺术（综合）

来源: University of Auckland * Department of Mathematics

【 摘 要 】

When a square matrix A, is diagonalizable, (for example,when A is Hermitian or has distinct eigenvalues), then Ancan 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 eigenvalueswith coefficients depending on n.In this case to a firstapproximation for large n, Anis 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.

【 授权许可】

Unknown