P-标准形与完全逆

摘要：

本文给出了结合环中元素的一种新的逆，即完全逆，作为某一方程组的唯一解。对于元素$a$，当且仅当$a$的Drazin逆存在时，该逆存在。我们还证明了通过在$-k$中插入$k$到方阵$A$的$\mathcal{P}$ -标准形中，我们得到了$A$的完全逆的$\mathcal{P}$ -标准形。

英文摘要：

This paper describes a new kind of inverse for elements in associative ring, that is the complete inverse, as the unique solution of a certain set of equations. This inverse exists for an element $a$ if and only if the Drazin inverse of $a$ exists. We also show that by plugging in $-k$ for $k$ in the $\mathcal{P}$-canonical form of a square matrix $A$, we get the $\mathcal{P}$-canonical form of the complete inverse of $A$.

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