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Wikipedia / Smith normal form

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 (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 moduless over a PID, and in particular for deducing the structure of a quotient of a free module. It is named after the Irish mathematician Henry John Stephen Smith.

Definition

matrix over a principal ideal domain units is

are unique up to multiplication by a unit and are called the elementary divisors, invariants, or invariant factors. They can be computed (up to multiplication by a unit) as

(called i-th determinant divisor) equals the greatest common divisor minors.

Example :.

Algorithm

), there are isomorphisms remain invertible matrices.

(these exist and are unique since any PID is also a unique factorization domain is also a Bézout domain, so it is a gcd domain satisfies a Bézout's identity.

.

Step I: Choosing a pivot

.

.

.

Step II: Improving the pivot

If there is an entry at position (k,jt, we know by the Bézout property that there exist σ, τ in R such that

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 (which divisions are possible by the definition of β) one has

so that the matrix

is invertible, with inverse

Now L 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,jt) (and due to our choice of α and γ it also has an entry 0 at position (k,jt; therefore repeating these steps must eventually terminate. One ends up with a matrix having an entry at position (t,jt) that divides all entries in column jt.

Step III: Eliminating entries

Finally, adding appropriate multiples of row t, it can be achieved that all entries in column jt except for that at position (t,jt) 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,jt) 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.

, and so the process must eventually stop after some number of iterations, leading to a matrix where the entry at position (t,jt) 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,jt) 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

are non-zero, and every other entry is zero.

.

in column i as in StepII; finally proceed as in Step, it is divisible by β.

submatrix), whence that operation does diminish (by moving prime factors to the right) the value of

as desired.

-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.

Run-time complexity

The Smith Normal Form of an N-by-N matrix A. If the matrix is sparse, the computation is typically much faster.

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).

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

References