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In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
(PID). The Smith normal form of a matrix is
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
, and can be obtained from the original matrix by multiplying on the left and right by
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
. It is named after the Irish mathematician Henry John Stephen Smith.


Definition

Let ''A'' be a nonzero ''m''×''n'' matrix over a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
''R''. There exist invertible m \times m and n \times n-matrices ''S, T'' (with coefficients in ''R'') such that the product ''S A T'' is \begin \alpha_1 & 0 & 0 & & \cdots & & 0 \\ 0 & \alpha_2 & 0 & & \cdots & & 0 \\ 0 & 0 & \ddots & & & & 0\\ \vdots & & & \alpha_r & & & \vdots \\ & & & & 0 & & \\ & & & & & \ddots & \\ 0 & & & \cdots & & & 0 \end. and the diagonal elements \alpha_i satisfy \alpha_i \mid \alpha_ for all 1 \le i < r. This is the Smith normal form of the matrix ''A''. The elements \alpha_i are unique
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
multiplication by a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
and are called the ''elementary divisors'', ''invariants'', or ''invariant factors''. They can be computed (up to multiplication by a unit) as : \alpha_i = \frac, where d_i(A) (called ''i''-th ''determinant divisor'') equals the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of the determinants of all i\times i minors of the matrix ''A'' and d_0(A):=1.


Algorithm

The first goal is to find invertible square matrices S and T such that the product S A T is diagonal. This is the hardest part of the algorithm. Once diagonality is achieved, it becomes relatively easy to put the matrix into Smith normal form. Phrased more abstractly, the goal is to show that, thinking of A as a map from R^n (the free R-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
of rank n) to R^m (the free R-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
of rank m), there are isomorphisms S:R^m \to R^m and T:R^n \to R^n such that S \cdot A \cdot T has the simple form of a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
. The matrices S and T can be found by starting out with identity matrices of the appropriate size, and modifying S each time a row operation is performed on A in the algorithm by the corresponding column operation (for example, if row i is added to row j of A, then column j should be subtracted from column i of S to retain the product invariant), and similarly modifying T for each column operation performed. Since row operations are left-multiplications and column operations are right-multiplications, this preserves the invariant A'=S'\cdot A\cdot T' where A',S',T' denote current values and A denotes the original matrix; eventually the matrices in this invariant become diagonal. Only invertible row and column operations are performed, which ensures that S and T remain invertible matrices. For a \in R\setminus \, write \delta(a) for the number of prime factors of a (these exist and are unique since any PID is also a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
). In particular, R is also a
Bézout domain In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every fini ...
, so it is a gcd domain and the gcd of any two elements satisfies a
Bézout's identity In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem: Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
. To put a matrix into Smith normal form, one can repeatedly apply the following, where t loops from 1 to m.


Step I: Choosing a pivot

Choose j_t to be the smallest column index of A with a non-zero entry, starting the search at column index j_+1 if t> 1. We wish to have a_\neq0; if this is the case this step is complete, otherwise there is by assumption some k with a_ \neq 0, and we can exchange rows t and k, thereby obtaining a_\neq0. Our chosen pivot is now at position (t, j_t).


Step II: Improving the pivot

If there is an entry at position (''k'',''j''''t'') such that a_ \nmid a_, then, letting \beta =\gcd\left(a_, a_\right), we know by the Bézout property that there exist σ, τ in ''R'' such that : a_ \cdot \sigma + a_ \cdot \tau=\beta. By left-multiplication with an appropriate invertible matrix ''L'', it can be achieved that row ''t'' of the matrix product is the sum of σ times the original row ''t'' and τ times the original row ''k'', that row ''k'' of the product is another linear combination of those original rows, and that all other rows are unchanged. Explicitly, if σ and τ satisfy the above equation, then for \alpha=a_/\beta and \gamma=a_/\beta (which divisions are possible by the definition of β) one has : \sigma\cdot \alpha + \tau \cdot \gamma=1, so that the matrix : L_0= \begin \sigma & \tau \\ -\gamma & \alpha \\ \end is invertible, with inverse : \begin \alpha & -\tau \\ \gamma & \sigma \\ \end . Now ''L'' can be obtained by fitting L_0 into rows and columns ''t'' and ''k'' of the identity matrix. By construction the matrix obtained after left-multiplying by ''L'' has entry β at position (''t'',''j''''t'') (and due to our choice of α and γ it also has an entry 0 at position (''k'',''j''''t''), which is useful though not essential for the algorithm). This new entry β divides the entry a_ that was there before, and so in particular \delta(\beta) < \delta(a_); therefore repeating these steps must eventually terminate. One ends up with a matrix having an entry at position (''t'',''j''''t'') that divides all entries in column ''j''''t''.


Step III: Eliminating entries

Finally, adding appropriate multiples of row ''t'', it can be achieved that all entries in column ''j''''t'' except for that at position (''t'',''j''''t'') are zero. This can be achieved by left-multiplication with an appropriate matrix. However, to make the matrix fully diagonal we need to eliminate nonzero entries on the row of position (''t'',''j''''t'') as well. This can be achieved by repeating the steps in Step II for columns instead of rows, and using multiplication on the right by the transpose of the obtained matrix ''L''. In general this will result in the zero entries from the prior application of Step III becoming nonzero again. However, notice that each application of Step II for either rows or columns must continue to reduce the value of \delta(a_), and so the process must eventually stop after some number of iterations, leading to a matrix where the entry at position (''t'',''j''''t'') is the only non-zero entry in both its row and column. At this point, only the block of ''A'' to the lower right of (''t'',''j''''t'') needs to be diagonalized, and conceptually the algorithm can be applied recursively, treating this block as a separate matrix. In other words, we can increment ''t'' by one and go back to Step I.


Final step

Applying the steps described above to the remaining non-zero columns of the resulting matrix (if any), we get an m \times n-matrix with column indices j_1 < \ldots < j_r where r \le \min(m,n). The matrix entries (l,j_l) are non-zero, and every other entry is zero. Now we can move the null columns of this matrix to the right, so that the nonzero entries are on positions (i,i) for 1 \le i\le r. For short, set \alpha_i for the element at position (i,i). The condition of divisibility of diagonal entries might not be satisfied. For any index i for which \alpha_i\nmid\alpha_, one can repair this shortcoming by operations on rows and columns i and i+1 only: first add column i+1 to column i to get an entry \alpha_ in column ''i'' without disturbing the entry \alpha_i at position (i,i), and then apply a row operation to make the entry at position (i,i) equal to \beta=\gcd(\alpha_i,\alpha_) as in Step II; finally proceed as in Step III to make the matrix diagonal again. Since the new entry at position (i+1,i+1) is a linear combination of the original \alpha_i,\alpha_, it is divisible by β. The value \delta(\alpha_1)+\cdots+\delta(\alpha_r) does not change by the above operation (it is δ of the determinant of the upper r\times r submatrix), whence that operation does diminish (by moving prime factors to the right) the value of :\sum_^r(r-j)\delta(\alpha_j). So after finitely many applications of this operation no further application is possible, which means that we have obtained \alpha_1\mid\alpha_2\mid\cdots\mid\alpha_r as desired. Since all row and column manipulations involved in the process are invertible, this shows that there exist invertible m \times m and n \times n-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 Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
of a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
when the chain modules of the chain complex are finitely generated. For instance, in
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, it can be used to compute the homology of a finite
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
or
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
over the integers, because the boundary maps in such a complex are just integer matrices. It can also be used to determine the
invariant factor The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then :M\cong R^r\o ...
s that occur in the
structure theorem for finitely generated modules over a principal ideal domain In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitel ...
, which includes the
fundamental theorem of finitely generated abelian groups In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
. The Smith normal form is also used in
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
to compute transmission and blocking zeros of a
transfer function matrix In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) systems to multiple-input and multiple-output ( ...
.


Example

As an example, we will find the Smith normal form of the following matrix over the integers. : \begin 2 & 4 & 4 \\ -6 & 6 & 12 \\ 10 & 4 & 16 \end The following matrices are the intermediate steps as the algorithm is applied to the above matrix. : \to \begin 2 & 0 & 0 \\ -6 & 18 & 24 \\ 10 & -16 & -4 \end \to \begin 2 & 0 & 0 \\ 0 & 18 & 24 \\ 0 & -16 & -4 \end : \to \begin 2 & 0 & 0 \\ 0 & 2 & 20 \\ 0 & -16 & -4 \end \to \begin 2 & 0 & 0 \\ 0 & 2 & 20 \\ 0 & 0 & 156 \end : \to \begin 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 156 \end So the Smith normal form is : \begin 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 156 \end 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 xI-A and xI-B have the same Smith normal form. For example, with : \begin A & =\begin 1 & 2 \\ 0 & 1 \end, & & \mbox(xI-A) =\begin 1 & 0 \\ 0 & (x-1)^2 \end \\ B & =\begin 3 & -4 \\ 1 & -1 \end, & & \mbox(xI-B) =\begin 1 & 0 \\ 0 & (x-1)^2 \end \\ C & =\begin 1 & 0 \\ 1 & 2 \end, & & \mbox(xI-C) =\begin 1 & 0 \\ 0 & (x-1)(x-2) \end. \end ''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 In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
*
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
*
Elementary divisors In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then '' ...
*
Frobenius normal form In linear algebra, the Frobenius normal form or rational canonical form of a square matrix ''A'' with entries in a field ''F'' is a canonical form for matrices obtained by conjugation by invertible matrices over ''F''. The form reflects a minimal ...
(also called Rational canonical form) *
Hermite normal form In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R''n'', the H ...
*
Invariant factor The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then :M\cong R^r\o ...
*
Structure theorem for finitely generated modules over a principal ideal domain In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitel ...
*
Singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...


Notes


References

* Reprinted (pp
367–409
i
''The Collected Mathematical Papers of Henry John Stephen Smith'', Vol. I
edited by J. W. L. Glaisher. Oxford: Clarendon Press (1894), ''xcv''+603 pp. * * {{PlanetMath , urlname=ExampleOfSmithNormalForm , title=Example of Smith normal form * 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
Matrix theory Matrix normal forms