矩阵的秩与其非零特征值个数相等的条件

The Equality Conditions between the Rank of a Matrix and Its Numbers of Nonzero Eigenvalue

证明了n阶方阵A的秩r(A)与其非零特征值个数μ(A)之间的关系:r(A)≥μ(A).得出了矩阵A可逆和矩阵A可对角化是r(A)=μ(A)的两个充分条件;矩阵A没有形如xm(m2)的初等因子是r(A)=μ(A)的充分必要条件.

The relationship between the rank of matrix and its number of nonzero eigenvalue is proved,that is r（A）≥u（A）.Following conclusion are educed,the two sufficient condition for r（A）=u（A） are that matrix A is invertible and matrix A is diagonalizable,the Necessary and Sufficient Conditions for r（A）=u（A） is that there is not the elementary factor whose form is xm（m≥2） in elementary factors of matrix A.