computes the Smith normal form of the matrix m of univariate polynomials.
computes the Smith normal form for a matrix of polynomials in the variable x.


PolynomialSmithReduce
computes the Smith normal form of the matrix m of univariate polynomials.
computes the Smith normal form for a matrix of polynomials in the variable x.
Details

- PolynomialSmithReduce is the polynomial analog of SmithReduce.
- The polynomial Smith reduction of a matrix m of univariate polynomials gives a a matrix s that is the Smith normal form of m.
- Diagonal elements of PolynomialSmithReduce[m] are constant multiples of the corresponding elements of PolynomialSmithDecomposition[m][[2]].
- A matrix is said to be Smith reduced, or in Smith normal form, if it is diagonal and each entry divides the subsequent entry.
- The matrix m need not be square and need not have full rank.
- The Smith form is canonical, that is, unique, up to multiplying diagonal elements by constants. Typically, these are normalized either by having pivots be monic or, if the base field is the rationals, having all polynomial coefficients be integers.
- Applications of PolynomialSmithReduce include finding polynomial solutions to polynomial systems, computing matrix minimal polynomials and finding the rational canonical form of a matrix.
Examples
open all close allBasic Examples (2)
Scope (4)
Generalizations & Extensions (1)
When a matrix has rational function entries, PolynomialSmithReduce will compute the Smith–McMillan form, where elements of s have a denominator:
Options (1)
Modulus (1)
PolynomialSmithReduce can work modulo a prime:
Applications (1)
Minimal Polynomial (1)
The last element in the diagonal of the Smith normal form of a characteristic matrix gives the minimal polynomial:
Compare to MatrixMinimalPolynomial:
Properties & Relations (3)
The second element of the Smith decomposition is a matrix with diagonal elements that are scalar multiples of the Smith normal form diagonal elements given by PolynomialSmithReduce.
Compute the Smith reduction of a 2×2 polynomial matrix:
Compute the decomposition and extract the second part:
The last element in the diagonal of the Smith normal form of a characteristic matrix gives the minimal polynomial:
Compare to MatrixMinimalPolynomial:
The polynomial Smith reduction can be used to compute the rational canonical form of a matrix; this is the second element in a Frobenius decomposition.
Compute the Smith form of a characteristic matrix and extract the diagonal:
Use this to create companion matrix blocks:
Use SparseArray and Band to reconstruct the companion matrix for m as a diagonal block matrix:
Related Guides
History
Text
Wolfram Research (2025), PolynomialSmithReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialSmithReduce.html.
CMS
Wolfram Language. 2025. "PolynomialSmithReduce." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PolynomialSmithReduce.html.
APA
Wolfram Language. (2025). PolynomialSmithReduce. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialSmithReduce.html
BibTeX
@misc{reference.wolfram_2025_polynomialsmithreduce, author="Wolfram Research", title="{PolynomialSmithReduce}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialSmithReduce.html}", note=[Accessed: 04-August-2025]}
BibLaTeX
@online{reference.wolfram_2025_polynomialsmithreduce, organization={Wolfram Research}, title={PolynomialSmithReduce}, year={2025}, url={https://reference.wolfram.com/language/ref/PolynomialSmithReduce.html}, note=[Accessed: 04-August-2025]}