### 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) |
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Pages (from-to) | 261-270 |

Number of pages | 10 |

Journal | Linear and Multilinear Algebra |

Volume | 21 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1 1987 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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*Linear and Multilinear Algebra*, vol. 21, no. 3, pp. 261-270. https://doi.org/10.1080/03081088708817800

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - On The Sum Of Powers Of Matrices

AU - Vaserstein, Leonid N.

PY - 1987/11/1

Y1 - 1987/11/1

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).

UR - http://www.scopus.com/inward/record.url?scp=33646854982&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646854982&partnerID=8YFLogxK

U2 - 10.1080/03081088708817800

DO - 10.1080/03081088708817800

M3 - Article

AN - SCOPUS:33646854982

VL - 21

SP - 261

EP - 270

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 3

ER -