# 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 language English (US) 261-270 10 Linear and Multilinear Algebra 21 3 https://doi.org/10.1080/03081088708817800 Published - Nov 1 1987

### Fingerprint

Sums of Powers
Sum of squares
Ring
Algebraic number Field
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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title = "On The Sum Of Powers Of Matrices",
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).",
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In: Linear and Multilinear Algebra, Vol. 21, No. 3, 01.11.1987, p. 261-270.

Research output: Contribution to journalArticle

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N2 - 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).

AB - 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).

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