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SmithReduce

SmithReduce

gives the Smith normal form of an integer matrix .

Details

The result is a diagonal matrix with each diagonal entry dividing the next one.
This result is the second element of the result given by SmithDecomposition.

Examples

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Basic Examples (1)

Give the Smith reduction of m into a diagonal matrix r:

Each entry on the diagonal of r divides the successor:

Scope (12)

Basic Uses (8)

Find the Smith reduction of an integer square matrix:

Smith reduction of a singular matrix:

The number of nonzero rows in r is equal to the rank of m:

Smith reduction for a rectangular matrix:

Use SmithReduce with a rational matrix:

Smith reduction for a Gaussian integer matrix:

Smith reduction for a Gaussian rational matrix:

Smith reduction works only with exact matrices:

The Smith reduction for a large, integer matrix is computed efficiently:

Special Matrices (4)

Find the Smith reduction for sparse matrices:

Smith reduction of structured matrices:

Smith reduction of an IdentityMatrix is simply an identity matrix:

Smith reduction of HilbertMatrix:

Applications (2)

The Smith reduction gives a canonical representation for finitely generated Abelian groups. Consider the quotient group of integer 2‐vectors modulo the relations and . That is to say, consider two points in the plane equivalent if they differ by integer multiples of the vectors and . Compute the Smith reduction of the matrix whose rows are the :

From , it can be seen that the structure of the group is , or simply , as is the trivial group:

Use SmithReduce to compute the extended GCD of two integers and :

Let be the column matrix with entries and :

The Smith reduction gives the GCD as well as a zero entry:

Compare to GCD:

Properties & Relations (6)

The Smith reduction is a diagonal, rectangular matrix of the same dimensions as :

Each diagonal entry in divides its successor:

The Smith reduction is the second matrix in the result of SmithDecomposition:

The matrix has integer values if the input matrix is integer:

If the input matrix has noninteger rational elements, so will the matrix:

For a rational-valued input, the diagonal entries of the matrix are non-negative:

For complex-valued input, the real and imaginary parts of the entries are non-negative:

The Smith reduction can often be found by two iterations of HermiteReduce:

The first use of HermiteReduce gives an upper triangular matrix:

Applying HermiteReduce to the Transpose of the triangular matrix gives :

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