Norms and inequalities for condition numbers
Norms and inequalities for condition numbers
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条件数的范数和不等式
DOI：
10.2140/pjm.1965.15.241
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发表时间：
1965
影响因子：
0.6
通讯作者：
I. Olkin
中科院分区：
文献类型：
--
作者：
A. W. Marshall;I. Olkin
摘要：非奇异矩阵A的条件数csub phi定义为：csub phi（A）= phi（A）phi（A上标-1）其中phi通常是范数。这是由J. D。Riley证明，如果A是正定的，则c sub phi（A + kI）=或0且phi squared（A）是AA* 的最大特征值或phi squared（A）= Tr AA*。本文较一般地证明了当V-U是正定的且φ满足φ（U）=或<φ（V）时，以及当A，B是正定的且满足c sub φ（A）=或< c sub φ（B）时，c sub φ（A + B）=或< c sub φ（B）.并得到了一些相关的不等式。正如Riley所建议的，当A是正定但病态的时，这些结果对于求解线性方程组Ax = d可能是有实际意义的。（作者）
Abstract : The condition number c sub phi of a nonsingular matrix A is defined by c sub phi (A) = phi (A) phi (A superscript -1) where ordinarily phi is a norm. It was shown by J. D. Riley that if A is positive definite, c sub phi (A + kI) = or 0 and phi squared (A) is the maximum eigenvalue of AA* or phi squared (A) = Tr AA*. In this paper it is shown more generally that c sub phi (A + B) = or < c sub phi (B) when phi satisfies phi (U) = or < phi (V) if V-U is positive definite and when A,B are positive definite satisfying c sub phi (A) = or < c sub phi (B). Some related inequalities are also obtained. As suggested by Riley, these results may be of practical use in solving a system Ax = d of linear equations when A is positive definite but ill-conditioned. (Author)