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Smith Normal Form

Let A be an n×n matrix over a field F. Using the three elementary row and column operations over elements in the field, the n×n matrix xI-A with entries from the principal ideal domain F[x] (where I is the identity matrix) can be put into the diagonal form

[1 0 ... 0 0 0 0 0; 0 1 ... 0 0 0 0 0; | ... ... ... ... ... ... |; 0 0 0 1 0 0 0 0; 0 0 0 0 a_1(x) 0 0 0; 0 0 0 0 0 a_2(x) 0 0; | ... ... ... ... ... ... |; 0 0 0 0 0 0 0 a_m(x)],

where a_1(x), a_2(x), ..., a_m(x) are monic nonzero elements of F[x] with degrees at least one and satisfying a_1(x)|a_2(x)|...|a_m(x), where f|g|h|... means f divides g, which in turn divides h, and so on (Dummit and Foote 1998, pp. 390-391 and 414). This form is known as Smith normal form, and the elements a_i(x) are called the invariant factors of A.

See also

Hermite Normal Form, Normal Form

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References

Ayres, F. Jr. "Smith Normal Form." Ch. 24 in Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, pp. 188-195, 1962.Dumas, J.-G.; Saunders, B. D.; and Villard, G. "On Efficient Sparse Integer Matrix Smith Normal Form Computations." J. Symb. Comput. 32, 71-100, 2001.Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998.Giesbrecht, M. "Fast Computation of the Smith Form of a Sparse Integer Matrix." Comput. Complexity 10, 41-69, 2001. Pascoletti, A. "Smith Normal Forms." http://library.wolfram.com/infocenter/MathSource/7081/.

Referenced on Wolfram|Alpha

Smith Normal Form

Cite this as:

Weisstein, Eric W. "Smith Normal Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmithNormalForm.html

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