Efficient characterization for I{2} and M{2}
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摘要: Stewart 给出了一个矩阵{2}-逆集合M{2}的刻划公式。 但公式.中含有多余的任意参数. 按Ben-Israel 的说法, 它不是一个有效刻划. 利用方阵的满秩分解, 本文定理 2..1 和2.2为I{2}的一个真子集B剔除了Stewart公式.中的多余任意参数, 得到了B的有效刻划公式;. 还证明了 I{2}是其有限个子集的并集, 其中每个子集与B等距同构. 由此可分别建立I{2}, I_{2}, M{2}和 M{2} 的有效刻划公式. 算法2..1则可用于无重复地计算 I{2}的每个元素.
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关键词:
- 广义逆矩阵类的有效刻划 /
- 矩阵的满秩分解 /
- 并集 /
- 集合的等距同构;
Abstract: An important characterization formula for M{2} was given by Stewart where M 2 Cmn. But this formula contains redundant arbitrary parameters, and therefore is nonefficient. This paper, by using the matrix full rank decomposition, showed that for a proper subset of I{2}s, which is denoted as B1, the redundant arbitrary parameters in Stewarts formula can be eliminated, and I{2}s is a union set of its certain subsets, and each of the subsets is 2-norm isometry with B1. Finally, the efficient characterization fonmulas for I{2}s, I{2} and M{2} are obtained respectively. An algorithm was provided that can be used to compute any element of I{2}s, and avoid the repeat computation work for each element of I{2}s. -
[1] BEN-ISRAEl A, GREVIllE T N E. Generalized Inverse: Theory and Applications [M]. 2nd ed., New York: Spring Verlag, 2003.DONG Z Q, YANG H. Characteristics of {2}-generalized inverses and some problems on partitioned matrices [J]. Pure Appl Math (Chinese), 1998, 14: 87-92.GOLUB G H, VAN LOAN C F. Matrix Computations [M]. 3rd ed. [s.l.]: Johns Hopkins University Press, 1996.HORN R, JOHNSON R. Matrix Analysis [M]. Cambridge: Cambridge University Press, 1990.SEWART G W. Projectors and generalized inverses [R]. Technical Report, TNN-97. [s.l.]: University of Texas at Austin Computation Center, 1969. -
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