nLab Smith normal form



Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts






(matrices over principal ideal domains equivalent to Smith normal form)

For RR a commutative ring which is a principal ideal domain (for instance R=R = \mathbb{Z} the integers), every matrix AMat n×m(R)A \in Mat_{n \times m}(R) with entries in RR is matrix equivalent to a diagonal matrix filled up with zeros:

There exist invertible matrices PMat n×n(R)P \in Mat_{n \times n}(R) and QMat m×m(R)Q \in Mat_{m \times m}(R) such that the product matrix PAQP A Q is of the following form:

PAQ= P A Q \;=\;

such that, moreover, each a ia_i divides a i+1a_{i+1}.


The results is named after

  • H.J.S. Smith, On systems of linear indeterminate equations and congruences Philos. Trans. Royal Soc. London cli, 293-326. Reprinted in The Collected Mathematical Papers of Henry John Stephen Smith, Volume 1. New York: Chelsea (1965)

Lecture notes include

  • Patrick Morandi, The Smith Normal Form of a Matrix, 2005 (pdf)

  • Sam Evans, Smith normal form over the integers (pdf)

  • Bill Casselman, Hermite and Smith forms, 2011 (pdf)

  • George Havas, Leon Sterling, Integer matrices and abelian groups (pdf, doi:10.1007/3-540-09519-5_94)

  • George Havas, Bohdan Majewski, Integer matrices and diagonalization, J. Symbolic Computation (1997) 24, 399-408 (pdf)

See also

Last revised on September 23, 2018 at 12:54:18. See the history of this page for a list of all contributions to it.