NationMaster - Encyclopedia: Smith normal form

The Smith normal form is a normal form that can be defined for any matrix (not necessarily square) with entries in a PID. The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible 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. The term normal form is used in a variety of contexts. ... PID stands for a number of things, including: Principal ideal domain, in abstract algebra, an integral domain in which every ideal is principal Proportional-Integral-Derivative controller, a feedback loop component in control theory and industrial control applications Pelvic inflammatory disease, an infection of the female uterus, fallopian tubes, and... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... PID stands for a number of things, including: Principal ideal domain, in abstract algebra, an integral domain in which every ideal is principal Proportional-Integral-Derivative controller, a feedback loop component in control theory and industrial control applications Pelvic inflammatory disease, an infection of the female uterus, fallopian tubes, and... In mathematics, a free module is a module having a free basis. ...

Let A be a nonzero m×n matrix over a principal ideal domain R, and for a in R {0}, write δ(a) for the number of prime factors of a (these exist and are unique since any PID is also a unique factorization domain). In particular, R is also a Bézout domain, so it is a gcd domain and the gcd of any two elements satisfies a Bézout's identity. In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ... In mathematics, a BÃ©zout domain, named after Ã‰tienne BÃ©zout, is an integral domain in which every finitely generated ideal is principal. ... In number theory, BÃ©zouts identity, named after Ã‰tienne BÃ©zout, is a linear diophantine equation. ...

1 Algorithm
2 Results
3 Applications
4 Example
5 External links

Algorithm

Our goal will to be find invertible square matrices S and T such that the product S A T is diagonal. This is the hardest part of the algorithm and once we have achieved diagonality it becomes relatively easy to put the matrix in Smith normal form. (Note that invertibility of a matrix with entries in R is the same as saying that its determinant is a unit.) Phrased more abstractly, the goal is to show that, thinking of A as a map from $R n$ (the free R-module of rank n) onto $R m$ (the free R-module of rank m), there are isomorphisms $S:R^m to R^m$ and $T:R^n to R^n$ such that $S circ A circ T$ has the simple form of a diagonal matrix. The matrices S and T will be found by repeatedly applying elementary transformations that replace a row (column) with a linear combination of itself and another row (column). The word unit means any of several things: Unit of measurement, a fundamental quantity of measurement Units (computer program), a popular program that does unit conversion Units of energy, the units for energy measurements Units conversion by the Factor-label method Functional unit, a component of a computer system such... A module is a self-contained component of a system, which has a well-defined interface to the other components; something is modular if it includes or uses modules which can be interchanged as units without disassembly of the module. ... A module is a self-contained component of a system, which has a well-defined interface to the other components; something is modular if it includes or uses modules which can be interchanged as units without disassembly of the module. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

Set t = 1 and choose j_t to be the smallest column index of A with a non-zero entry. One can repeatedly apply the following to put a matrix into Smith normal form.

Case I

If $a_{t,j_t}=0$ and $a_{k,j_t} neq 0$ , exchange rows 1 and k.

Case II

If there is an entry at position (k,j_t) such that $a_{t,j_t} nmid a_{k,j_t}$ , then, letting $beta =gcdleft(a_{t,j_t}, a_{k,j_t}right)$ , we know by the Bézout property that there exist σ, τ in R such that

$sigmacdot a_{t,j_t}-tau cdot a_{k,j_t}=beta.$

By left-multiplication with an appropriate matrix L it can be achieved that row t of the matrix product is the sum of row t multiplied by σ and row k multiplied by -τ. (If σ and τ satisfy the above equation, they must be relatively prime; so there exist α and γ such that In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...

$sigmacdot alpha+tau cdot gamma=1,$

or in other words, the determinant of the matrix In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

$begin{pmatrix} sigma & -tau gamma & alpha end{pmatrix}$

equals one. L can be obtained by fitting this matrix into the diagonal of the identity matrix at the appropriate positions, depending on the value of t and k. That L has determinant one guarantees that L is invertible over R.) After left-multiplying by L we get β at position (t,j_t), where $delta(beta) < delta(a_{t,j_t})$ and β divides $a_{t,j_t}$ . Repeating these steps, one ends up with a matrix having an entry at position (t,j_t) that divides all entries in column j_t.

Case III

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 Case II for columns instead of rows, and using multiplication on the right. In general this will result in the zero entries from the prior application of Case II becoming nonzero again.

However, notice that the ideals generated by the elements at position (t,j_t) form an ascending chain, because entries from a later step always divide entries from a previous step. Therefore, since R is a Noetherian ring (it is a PID), the ideals eventually become stationary and do not change. This means that at some stage after Case II has been applied, the entry at (t,j_t) will divide all nonzero row or column entries before applying any more steps in Case II. Then we can eliminate entries in the row or column with nonzero entries while preserving the zeros in the already-zero row or column. At this point, only the block of A to the lower right of (t,j_t) needs to be diagonalized, and the algorithm can be applied recursively, treating this block as a separate matrix. Ideals are principles or values that one actively pursues as goals. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... PID stands for a number of things, including: Principal ideal domain, in abstract algebra, an integral domain in which every ideal is principal Proportional-Integral-Derivative controller, a feedback loop component in control theory and industrial control applications Pelvic inflammatory disease, an infection of the female uterus, fallopian tubes, and...

Results

It has been suggested that Elementary divisors be merged into this article or section. (Discuss)

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 $r le min(m,n)$ , each of which satisfies the following: Image File history File links Please see the file description page for further information. ... It has been suggested that this article or section be merged into Smith normal form. ...

the entry at position $(l, j l)$ is non-zero;
all entries below and above position $(l, j l)$ as well as entries left of $(l, j l)$ are zero.

Furthermore, all rows below the $r$ -th row are zero.

This is a version of the Gauss algorithm for principal ideal domains which is usually described only for commutative fields. In mathematics, Gaussian elimination or Gauss-Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... This article presents the essential definitions. ...

Now we can re-order the columns of this matrix so that elements on positions $(i, i)$ for $1 le ile r$ are nonzero and $delta(a_{ii}) le delta(a_{i+1,i+1})$ for $1 le i < r$ ; and all columns right of the $r$ -th column (if present) are zero. For short set $α i$ for the element at position $(i, i)$ . $δ$ has non-negative integer values; so $δ(α 1) = 0$ is equivalent to $α 1$ being a unit of $R$ .

$δ(α i) = δ(α i + 1)$ can either happen if $α i$ and $α i + 1$ differ by a unit factor, or if they are relative prime. In the latter case one can add column $i + 1$ to column $i$ (which doesn't change $α i)$ and then apply appropriate row manipulations to get $α i = 1$ . And for $δ(α i) < δ(α i + 1)$ and $alpha_i nmid alpha_{i+1}$ one can apply step (II) after adding column $i + 1$ to column $i$ .

This diminishes the minimum $δ$ -values for non-zero entries of the matrix, and by reordering columns etc. we end up with a matrix whose diagonal elements $α i$ satisfy $alpha_i mid alpha_{i+1};forall;1 le i < r$ .

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 is

$begin{pmatrix} alpha_1 & 0 & 0 & & cdots & & 0 0 & alpha_2 & 0 & & cdots & & 0 0 & 0 & ddots & & & & 0 vdots & & & alpha_r & & & vdots & & & & 0 & & & & & & & ddots & 0 & & & cdots & & & 0 end{pmatrix}$

This is the Smith normal form of the matrix. The elements $α i$ are unique up to associatedness and are called the elementary divisors, invariants, or invariant factors. Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In Linear Algebra if we have a matrix M in smith normal form as seen below. ...

Applications

The Smith normal form is useful for computing the homology of a chain complex when the elements of the chain complex are finitely generated. For instance, in topology, it can be used to compute the homology of a 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 prove the well known structure theorem about finitely generated modules over a PID. Homology is an important concept in several disciplines: Homology (anthropology) in archaeology and anthropology. ... In mathematics, in the field of homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Homology is an important concept in several disciplines: Homology (anthropology) in archaeology and anthropology. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... A module is a self-contained component of a system, which has a well-defined interface to the other components; something is modular if it is constructed so as to facilitate easy assembly, flexible arrangement, and/or repair of the components. ...

Example

As an example, we will find the Smith normal form of the following matrix over the integers.

$begin{pmatrix} 2 & 4 & 4 -6 & 6 & 12 10 & -4 & -16 end{pmatrix}$

The following matrices are the intermediate steps as the algorithm is applied to the above matrix.

$to begin{pmatrix} 2 & 0 & 0 -6 & 18 & 24 10 & -24& -36 end{pmatrix} to begin{pmatrix} 2 & 0 & 0 0 & 18 & 24 0 & -24& -36 end{pmatrix}$

$to begin{pmatrix} 2 & 0 & 0 0 & 18 & 24 0 & -6 & -12 end{pmatrix} to begin{pmatrix} 2 & 0 & 0 0 & 6 & 12 0 & 18 & 24 end{pmatrix}$

$to begin{pmatrix} 2 & 0 & 0 0 & 6 & 12 0 & 0 & -12 end{pmatrix} to begin{pmatrix} 2 & 0 & 0 0 & 6 & 0 0 & 0 & 12 end{pmatrix}$

So the Smith normal form is

$begin{pmatrix} 2 & 0 & 0 0 & 6 & 0 0 & 0 & 12 end{pmatrix}$

and the elementary divisors are 2, 6 and 12.

External links

Thomas Heye's GFDL Smith normal form article at PlanetMath
GFDL Example of Smith normal form at PlanetMath

Categories: Articles to be merged | Matrix theory

Results from FactBites:

Henry John Stephen Smith (190 words)
	The Smith normal form for matrices are named after him.
	He was born in Dublin, Ireland, the fourth child of John Smith, a barrister.
	Smith remained at Balliol, becoming a fellow (1850), then working as a tutor before being appointed Savilian Professor[?] of Geometry in 1861.

Bibliography (465 words)
	Smith normal form of dense integer matrices, fast algorithms into practice.
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More results at FactBites »

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