Equivalence of matrices over R[s, z] : a counter-example
Equivalence of matrices over R[s, z] : a counter-example
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DOI：
10.1080/00207178108922596
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发表时间：
1981-12
影响因子：
2.1
通讯作者：
M. G. Frost;C. Storey
中科院分区：
文献类型：
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作者：
M. G. Frost;C. Storey
Frost和Storey（1978）研究了lR [s， z]上矩阵等价的概念及其对此类矩阵的零的依赖。特别提出，除了必要条件外，矩阵在lR [s， z]上无零的条件也足以使它在lR [s， z]上等价于它在lR [s， z]上的Smith形式。然而，弗罗斯特（1979）在对这一结果所依据的材料的扩展说明中指出，由于结果的证明存在弱点，需要进一步调查。现在已经有可能提供一个反例来证明所提出的结果。这个反例依赖于另一个结果，这个结果是一个众所周知的关于lR上矩阵的结果的扩展[s]（参见，例如Rosenbrock和Storey 1970）。设A (z)和A'(z)是lR [z]上的n × n个矩阵。那么矩阵sl；当且仅当A (z)和A'(z)在lR [s， z]上相似时，-A (z)和sIn-A'(z)在lR [z]上相等。当且仅当存在一个n × n的单模矩阵H (z) / lR [z]满足
Frost and Storey (1978) investigated the concept of equivalence of matrices over lR [s, z] and its dependence on the zeros of such matrices. In particular it was proposed that, in addition to being necessary, the condition that a matrix over lR [s, z] has no zeros is also sufficient for it to be equivalent over lR [s, z] to its Smith form over lR [s, z]. However, Frost (1979) in an expanded account of the material on which this result was based pointed out that further investigation was needed as there existed a weakness in the proof of the result. It has now become possible to provide a counter-example to the result as proposed. This counter-example depends on another result which is an extension of a well-known result for matrices over lR [s](see, eg Rosenbrock and Storey 1970).Let A (z) and A'(z) be n x n matrices over lR [z]. Then the matrices sl;-A (z) and sIn-A'(z) are equivalent over lR [s, z] if and only if A (z) and A'(z) are similar over lR [z]. That is if and only if there exists a unimodular n x n matrix H (z) over lR [z] such that