摘要:
集合\,$\mathbb{A}$\,到集合\,$\mathbb{B}$\,上的一个一一映射\,$f$\,称为\,$\mathbb{B}$\,的一个有效刻画.本文提出的选逆象指标法\\\,(SIIIM)\,给出集\,${\mathbb{A}}_1=\{\alpha:\alpha=(I_s,\eta)^{\rmT}\in \mathbb{C}^{n\timess}_{s}\}$\,到象集\,${\mathbb{B}}_1=\{\beta:\beta=\alpha(\alpha^{\ast}\alpha)^{-1}\alpha^{\ast},\alpha\in{\mathbb{A}}_1\}$\,的一个有效刻画公式,并证明了\,${\mathbb{B}}_1$\,是\,$I\{2,3\}_s$\,的稠密子集，且\,$I\{2,3\}_s$\,的每个元素都与\linebreak
${\mathbb{B}}_1$\,的某个元素置换相似. 利用上述结果,分别建立了\,$I\{2,3\}$\,和长方阵广义逆矩阵类\,$M\{2,3\}$\linebreak的有效刻画公式.再利用等式\,$I\{2,3\}_s\!=\!I\{2,4\}_s\!=\!I\{2,3,4\}_s$,进一步获得了\,$M\{2,4\}$, $M\{2,3,4\}$\,的有效刻画公式.算法\,3.1\,可用于无重复地计算\,$I\{2,3\}_s$\,的任一个元素
Abstract:
In this paper, by {\it selecting inverse image index method}, an efficient characterization formula from set ${\mathbb{A}}_{1}=\{\alpha:\alpha=(I_s,\eta^{\rm T})^{\rm T}\in \Bbb C^{n\times s}_s\}$ onto set ${\mathbb{B}}_1=\{\beta:\beta=\alpha(\alpha^*\alpha)^{-1}\alpha^*, \alpha\in{\mathbb{A}}_1\}$ is given. Besides, it is shown that each element of $I\{2,3\}_s$ is permutation similar to an element of ${\mathbb{B}}_1$. Then efficient characterization formulas for $I\{2,3\}$ and $M\{2,3\}$ are obtained respectively. An interesting thing is ${\mathbb{B}}_1$ is a dense subset of $I\{2,3\}_s$. The fact that $I\{2,3\}_s=I\{2,4\}_s=I\{2,3,4\}_s$ enables us to obtain the efficient characterization formulas for $M\{2,4\}$ and $M\{2,3,4\}$ fluently. Algorithm 3.1 may be used to compute elements of $I\{2,3\}_s$ and to avoid the repeated computation work.
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