巧用初等矩阵解决线性代数中的四个重要问题

Using Elementary Matrices to Solve Four Important Problems in Linear Algebra

从初等矩阵的定义和性质出发,总结并整理了矩阵化为等价标准型、矩阵方程与矩阵逆的求解、实对称矩阵的相似对角化以及对称矩阵的合同对角化四类问题.利用初等变换可方便地得到上述矩阵求解问题中的变换矩阵,该求解方法也是初等变换与初等矩阵在线性代数中的一项重要应用.

We discuss four kinds of problems:transforming into equivalent standard form,solving matrix equations and inverse matrix,diagonalizing similarly or congruently a real symmetric matrix.The results and corresponding transformation matrices can be solved easily by using elementary transformations.