中文摘要：

设R是一个交换主理想整环（PID），A，B是两个R上的对称矩阵，讨论了A与B的算术距离与距离的关系，证明了A—B可合同对角化的充要条件是：A与B的距离等于它们的算术距离.

英文摘要：

Let R be a commutative principal ideal domain, A and B be two symmetric matrices over R. We discuss the relationship between the arithmetic distance and the distance of A and B, and prove that A - B is cogradient to a diagonal matrix if and only if the distance between A and B equals to their arithmetic distance.