## [Smith Normal Form](/smith-normal-form) ## [Smith Normal Form](https://mathlogic.quantumdictionary.io/smith-normal-form-0) [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic) - Natural & Formal Sciences - **Mathematics & Logic Dictionary** Definition A canonical diagonal form for an integer (or more generally PID) matrix achieved by left and right multiplication by unimodular matrices: there exist U, V unimodular over Z such that U A V = diag(d1, d2, ..., dr, 0, ...), where each di divides the next; the diagonal entries are the invariant factors of the associated module. ## [Smith Normal Form](https://puremath.quantumdictionary.io/smith-normal-form-1) [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics) - Mathematics & Logic - **Pure Mathematics Dictionary** Definition A canonical diagonal form for matrices over the integers (or more generally over a principal ideal domain) obtained by left and right multiplication by unimodular matrices, whose diagonal entries d1, d2, ... satisfy divisibility d1 | d2 | ... and encode the module structure of the cokernel. ## [Smith Normal Form](https://algebra.quantumdictionary.io/smith-normal-form-2) [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra) - Pure Mathematics - **Algebra Dictionary** Definition A canonical diagonal form obtainable for a matrix over a principal ideal domain (PID) via invertible row and column operations, producing diagonal entries d1,…,dr (possibly with trailing zeros) satisfying d1 | d2 | … | dr; the diagonal entries are unique up to multiplication by units and encode invariant factors of the corresponding module.