矩阵对角化中可逆矩阵的研究

Researches on Invertible Matrix in Diagonalizable Matrix

通过对矩阵的对角化研究,找出了在对角化过程中所取的可逆阵之间的内在联系,并分别从矩阵以及线性变换两个角度给出了可逆阵之间的关系。如果n阶方阵A可对角化,则存在可逆阵P,使P-1AP为对角阵。若取可逆阵Q,Q-1AQ也为对角阵,那么适当调整Q的列向量的次序后,调整后的P,Q的列向量之间存在线性关系,且列向量之间的线性变换的矩阵为准对角矩阵,该准对角矩阵的每个块矩阵的阶数等于A的某个特征值的重数,并举例说明了这一结论。

We find relations among invertible matrixs in diagonalization by studying of diagonalizable matrix.And we give relations of invertible matrixs from the view of matrix and linear transformation.An×n is diagonalizable if and only if there exists a invertible matrix P,such that the matrix P-1AP is a diagonal matrix.If Q is invertible,and Q-1AQ is a diagonal matrix,then there exists linear relations between column vectors of P,Q after we adjust the order of them.And the matrix of linear transition is a block diagonal matrix,and the order of each block matrix equals multiplicity of some eigenvalue.We also give an example to prove it.