In mathematics, the Smith normal form is a normal form that can be defined for any matrix (not necessarily square) 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.
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Famous quotes containing the words smith, normal and/or form:
“People of the same trade seldom meet together, even for merriment and diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices.”
—Adam Smith (1723–1790)
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—Bible: New Testament Jesus, in John, 20:17.
Spoken to Mary Magdalene, after Jesus has risen from the dead and made himself known to her. The words are best known in the Latin form in which they appear in the Vulgate: Noli me tangere.