On The Sum Of Powers Of Matrices

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

A theorem of Lagrange says that every natural number is the sum of 4 squares. M. Newman proved that every integral n by n matrix is the sum of 8 − (−1)n squares when n is at least 2. He asked to generalize this to the rings of integers of algebraic number fields. We show that an n by n matrix over a commutative ring R with 1 is the sum of squares if and only if its trace reduced modulo 2R is a square in the ring R/2R. If this is the case (and n is at least 2), then the matrix is the sum of 6 squares (5 squares would do when n is even). Moreover, we obtain a similar result for an arbitrary ring R with 1. Answering another question of M. Newman, we show that every integral n by n matrix is the sum of ten k-th powers for all sufficiently large n (depending on k).

Original languageEnglish (US)
Pages (from-to)261-270
Number of pages10
JournalLinear and Multilinear Algebra
Volume21
Issue number3
DOIs
StatePublished - Nov 1 1987

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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