Multiplicative perturbations of matrices and the generalized triple reverse order law for the Moore–Penrose inverse
Multiplicative perturbations of matrices and the generalized triple reverse order law for the Moore–Penrose inverse
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DOI：
10.1016/j.laa.2017.05.026
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发表时间：
2017-10
影响因子：
1.1
通讯作者：
Qingxiang Xu;Chuanning Song;Guo-rong Wang
中科院分区：
文献类型：
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作者：
Qingxiang Xu;Chuanning Song;Guo-rong Wang
对于矩阵a，设a†表示它的Moore-Penrose逆。调用矩阵M T的乘法扰动∈C M×n如果M = E T F⁎E∈F C M×M和∈C n×n。基于另一种表达式为M M = (E T T†)⋅T⋅(F T†)⁎,广义三重倒序法为获得Moore-Penrose逆M†= ((F T†)⁎)__⋅(Y Y†T Z Z†)L R−1 __⋅(E T T __),在(Y Y†T Z Z†)L R−1 __是一定的加权Moore-Penrose逆矩阵Y, Z, L和R相关联的三倍(T, E, F)。进一步证明了当（E T T†）†E T T†⋅T= T⋅（F T†T）†（F T†T）时，M†的加权Moore-Penrose逆确实可以被T†代替。在rank (M)= rank (T)或M是T的弱摄动的特殊情况下，也得到了M†的简化形式，以及M M†和M†M。
For a matrix A, let A† denote its Moore–Penrose inverse. A matrix M is called a multiplicative perturbation of T∈ C m× n if M= E T F⁎ for some E∈ C m× m and F∈ C n× n. Based on the alternative expression for M as M=(E T T†)⋅ T⋅(F T† T)⁎, the generalized triple reverse order law for the Moore–Penrose inverse is obtained as M†=((F T† T)⁎)†⋅(Y Y† T Z Z†) L R− 1†⋅(E T T†)†, where (Y Y† T Z Z†) L R− 1† is the weighted Moore–Penrose inverse for certain matrices Y, Z, L and R associated to the triple (T, E, F). Furthermore, it is proved that this weighted Moore–Penrose inverse in the resulting expression for M† can be really replaced with T† if (E T T†)† E T T†⋅ T= T⋅(F T† T)†(F T† T). In the special case that rank (M)= rank (T) or M is a weak perturbation of T, a simplified version of M†, as well as M M† and M† M, is also derived.