Integral matrices as diagonal quadratic forms

Abstract

In this paper, we investigate the conditions under which a diagonal quadratic form ${\sum }_{i=1}^m{a}_i{X}_i^2$ represents every $n\times n$ integral matrix, where $a_i$ ($1\le i\le m$) are integers. For $n=2$, we give a necessary and sufficient condition. Also we give some sufficient conditions for each $n\ge 2$ where $a_i$ ($1\le i\le m$) are pairwisely coprime.

5 pages

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