possible, which means that we have obtained as desired. Since all row and column manipulations involved in the process are invertible, this shows that there exist invertible and -matrices S, T so that the product S A T satisfies the definition of a Smith normal form. In particular, this shows that the Smith normal form exists, which was assumed without proof in the definition. Applications The Smith normal form is useful for computing the homology of a chain complex when the chain modules of the chain complex are finitely generated. For instance, in topology, it can be used to compute the homology of a finite simplicial complex or CW complex over the integers, because the boundary maps in such a complex are just integer matrices. It can also be used to determine the invariant factors that occur in the structure theorem for finitely generated modules over a principal ideal domain, which includes the fundamental theorem of finitely generated abelian groups. The Smith normal form is also used in control theory to compute transmission and blocking zeros of a transfer function matrix. Example As an example, we will find the Smith normal form of the following matrix over the integers. The following matrices are the intermediate steps as the algorithm is applied to the above matrix. So the Smith normal form is and the invariant factors are 2, 2 and 156. Similarity The Smith normal form can be used to determine whether or not matrices with entries over a common field are similar. Specifically two matrices A and B are similar if and only if the characteristic matrices and have the same Smith normal form. For example, with A and B are similar because the Smith normal form of their characteristic matrices match, but are not similar to C because the Smith normal form of the characteristic matrices do not match. See also Canonical form Diophantine equation Elementary divisors Frobenius normal form (also called Rational canonical form) Hermite normal form Invariant factor Structure theorem for finitely generated modules over a principal ideal domain Singular value decomposition Notes References Reprinted (pp. 367–409) in The Collected Mathematical Papers of Henry John Stephen Smith, Vol. I, edited by J. W. L. Glaisher. Oxford: Clarendon Press (1894), xcv+603 pp. K. R. Matthews, Smith normal form. MP274: Linear Algebra, Lecture Notes, University of Queensland, 1991. External links An animated example of computation of Smith normal form.