Let A be a matrix over ring Z/6Z. How to find an invertible matrix P and Q such that the matrix PAQ isa Smith Normal Form of A?
This does not make sense. See Wikipedia
sage: R = Zmod(6)
sage: R in Domains()
False
sage: M=matrix(R,[[4,3],[2,1]])
sage: M.smith_form()
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Traceback (most recent call last)
TypeError: Smith form only defined over Noetherian integral domainsIn fact, the error is misleading: it should say that the Smith form makes sense only over Principal ideal rings which are domains (I wrote it like that for emphasis: of course, I mean the Principal Ideal Domains). There is a similar structure theorem for modules over Dedekind domains, but now the torsion-free = projective which is free + fractional ideal. This probably is not implemented. At any rate, this is not Smith normal form (probably, this must be called the Steinitz normal form...).
Asked: 10 years ago
Seen: 3,268 times
Last updated: Aug 08 '14
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