# A verified algorithm for computing the Smith normal form of a matrix

 Title: A verified algorithm for computing the Smith normal form of a matrix Author: Jose Divasón Submission date: 2020-05-23 Abstract: This work presents a formal proof in Isabelle/HOL of an algorithm to transform a matrix into its Smith normal form, a canonical matrix form, in a general setting: the algorithm is parameterized by operations to prove its existence over elementary divisor rings, while execution is guaranteed over Euclidean domains. We also provide a formal proof on some results about the generality of this algorithm as well as the uniqueness of the Smith normal form. Since Isabelle/HOL does not feature dependent types, the development is carried out switching conveniently between two different existing libraries: the Hermite normal form (based on HOL Analysis) and the Jordan normal form AFP entries. This permits to reuse results from both developments and it is done by means of the lifting and transfer package together with the use of local type definitions. BibTeX: @article{Smith_Normal_Form-AFP, author = {Jose Divasón}, title = {A verified algorithm for computing the Smith normal form of a matrix}, journal = {Archive of Formal Proofs}, month = may, year = 2020, note = {\url{http://isa-afp.org/entries/Smith_Normal_Form.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: Berlekamp_Zassenhaus, Hermite, List-Index, Perron_Frobenius