“和差化积”在抽象矩阵运算中的应用

Applications of the Product to Sum in the Abstract Matrix

抽象矩阵的运算是线性代数考研题中常见的一种题型,但由于抽象矩阵的具体元素未知,所以只能综合运用矩阵的性质来计算。在矩阵乘积的运算中,|AB|=|A||B|,(AB)-1=B-1A-1等性质可以大大简化运算。但在矩阵的和差运算中,由于|A B||A||B|,(A B)-1 A-1 B-1,因此必须把和差转化为乘积,即"和差化积",从两个方面说明"和差化积"的应用。

In Linear Algebra,the computation of abstract matrix is important.For the abstract matrix,we must use the properties of matrix because the exact elements are unknown.In Matrix product,the properties of matrices can simplify operations,so we can translate the sum difference of the matrix into matrix product.In this paper,the author explains this application from two aspects.