Minimal properties of Moore-Penrose inverses
Minimal properties of Moore-Penrose inverses
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DOI：
10.1016/0024-3795(94)90322-0
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发表时间：
1994
影响因子：
1.1
通讯作者：
D. R. Jensen
中科院分区：
文献类型：
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作者：
D. R. Jensen
设Ax = y是相容的，设x 0 = Gy是满足（AG）′ =AG的最小范数解，设A+是A的Moore-Penrose逆.证明了对于包含酉不变矩阵范数的类Φ中的任意φ，φ（G）<$φ（A+）.利用条件数C φ（A，G）研究了系统Ax = y的条件化.证明了对任意φ∈ Φ，C φ（A，G）<$Cφ（A，A+）.此外，还用奇异值的形式给出了C φ（A，G）的界.并行结果发现当AandG是对称的，与应用程序的线性模型小于满秩。
LetAx=ybe consistent; letx0=Gybe any minimum-norm solution satisfying (AG)′ =AG; and letA+be the Moore-Penrose inverse ofA. It is shown that φ(G) ⩾ φ(A+) for any φ in a class Φ containing the unitarily invariant matrix norms. The conditioning of the systemAx=yis studied via condition numbersCφ(A, G). It is shown thatCφ(A, G) ⩾Cφ(A, A+) for every φ∈ Φ. Moreover, bounds onCφ(A, G) are given in terms of singular values. Parallel results are found whenAandGare symmetric, with applications to linear models of less than full rank.