On The Sum Of Powers Of Matrices

Research output: Contribution to journalArticle

9 Citations (Scopus)

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

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Sums of Powers
Sum of squares
Ring
Algebraic number Field
Question Answering
Natural number
Commutative Ring
Lagrange
Modulo
Trace
If and only if
Generalise
Integer
Arbitrary
Theorem

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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On The Sum Of Powers Of Matrices. / Vaserstein, Leonid N.

In: Linear and Multilinear Algebra, Vol. 21, No. 3, 01.11.1987, p. 261-270.

Research output: Contribution to journalArticle

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