对合矩阵与对角矩阵相似的一种证法

A Method for Proving the Similarity of Involutory Matrices and Diagonal Matrices

我们利用向量组的线性相关性以及分块矩阵的运算性质给出了下列命题的另一种有趣的证法：若n阶对合矩阵A满足条件秩（A＋In）=r，则A相似于对角矩阵diag{Ir，-In-r}．这种证法连同Schmidt标准正交化方法一起，还可以用来证明：当上述矩阵A是实对称（Hermite）矩阵时，A正交（酉）相似于对角矩阵diag{Ir,-n-r}．

In this paper, by using the linear dependency of a system of vectors and the operation properties of block matrices, we give another interesting method for proving the proposition that if A is an n × n involutory matrix with textrank （A ＋ In） = r , then A is similar to the diagonal matrix textdiag { Ir, - In - r } . This method together with the Sehmidt＇s orthogonormalization can also be used to prOve that A is orthogonal （unitary） similar to the diagonal matrix textdiag {Ir, - In - r], whenever the above matrix A is a real symmetry （Hermitian） matrix.