摘要
The well-known trace equality of similar matrices does not necessarily hold for matrices over non-commutative algebras and rings. An interesting question is to give conditions such that trace equality of similar matrices holds for matrices over a non-commutative algebra or ring. in this note, we show that for any two matrices A and B over a generalized quaternion algebra defined on an arbitrary field F of characteristic not equal to two, if A and B are similar and the main diagonal elements of A and B are in F, then their traces are equal.
众所周知,相似矩阵的迹相等对于非交换代数和环上的矩阵不一定成立,有趣的问题是给定一个条件使得相似矩阵的迹相等对于非交换代数或非交换环上的矩阵成立。本文对于特征不是2的任意域F上定义的广义四元数代数上的两个矩阵A和B,给出如果A和B相似并且它们的主对角线上的元素在F中,那么它们的迹相等。
