In mathematics, the Smith normal form (sometimes abbreviated SNF[1]) 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. It is named after the Irish mathematician Henry John Stephen Smith.
^Stanley, Richard P. (2016). "Smith normal form in combinatorics". Journal of Combinatorial Theory. Series A. 144: 476–495. arXiv:1602.00166. doi:10.1016/j.jcta.2016.06.013. S2CID 14400632.
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defines the characteristic polynomial). Note that this Smithnormalform is not a normalform of A itself; moreover it is not similar to XIn − A either...
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equivalent with multiplying M on the right by a unimodular matrix. The Smithnormalform of M is a matrix S = U M V , {\displaystyle S=UMV,} where U and V...
needed] The finitely presented case is solved by Smithnormalform, and hence frequently credited to (Smith 1861), though the finitely generated case is sometimes...
associatedness. The invariant factors of a matrix over a PID occur in the Smithnormalform and provide a means of computing the structure of a module from a...
existence part of the theorem is provided by any algorithm computing the Smithnormalform of a matrix of integers. Uniqueness follows from the fact that, for...
characteristic polynomial, eigenvalues (but not eigenvectors), Hermite normalform, and Smithnormalform.[citation needed] In his 1969 paper, where he proved the complexity...
application of the Smithform algorithm get filled-in even if one starts and ends with sparse matrices. Efficient and probabilistic Smithnormalform algorithms...
diagonal is equal to k {\displaystyle k} . This is a special case of the Smithnormalform, which generalizes this concept on vector spaces to free modules over...
generators), and put it in Smithnormalform. This yields the invariant factor decomposition, and the diagonal entries of Smithnormalform are the invariant factors...
decomposition Principal component analysis (PCA) Schmidt decomposition Smithnormalform Singular value Time series Two-dimensional singular-value decomposition...
linear Diophantine equations and abelian group with Hermite normalform and Smithnormalform. Before the 20th century, different types of eliminants were...
example of an additive relation. Mapping cone (homological algebra) Smithnormalform Chain complex Pairing Bourbaki, Ch. II, §1.14, remark 2. Matsumura...
programme), 2013–2016 Server Normal Format, a bitmap font format used by X Window Skilled nursing facility Smithnormalform Spent nuclear fuel This disambiguation...
[q]} ) for the arrangement and much information is contained in its Smithnormalform. In complex affine space (which is hard to visualize because even...
normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution...
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voiced by Christopher Corey Smith in The Matrix: Path of Neo (2005), with Ian Bliss and Gideon Emery playing his human form, Bane, in the films and Path...
societal values, ideologies and dominant narratives in the form of curriculum. The first public normal school in the United States was founded in Concord, Vermont...
in 2012. Smith is also the drummer of the hard rock supergroup Chickenfoot, formed in 2008, and of the all-instrumental outfit Chad Smith's Bombastic...
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