Integral matrices as diagonal quadratic forms

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 $n \geq 2$where$a_i$($1 \leq i \leq m$) are pairwisely coprime.