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Smith normal form

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Smith normal form

normal form for a matrix with values in a principal ideal domain

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About Smith normal form

In mathematics, the Smith normal form is a normal form that can be defined for any matrix with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a free module. It is named after the Irish mathematician Henry John Stephen Smith.

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