Decompose m into unimodular matrices u and v and a diagonal matrix r:

The determinants of u and v have absolute value 1:

Each entry on the diagonal of r divides the successor:

Confirm the decomposition:

The Smith decomposition can be used to solve linear equations over the integers. Consider with and as follows:

As , the equation implies :

Compare with the solution from Solve:

For an invertible , a solution exists if and only if the above process produces an integer result:

The Smith decomposition 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 decomposition 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:

Hence, every point on the plane is equivalent to one of the nine points . Consider the point . To find the canonical representative of , rotate into the diagonal basis using on the right and compute the modulus with respect to the diagonal entries of :

The point is equivalent to , so it should map to the same canonical representative, and it does:

Thus, every integer point in the plane can be reduced to a canonical representative. Conversely, the matrix can be viewed as mapping the canonical basis to the starting basis. As the lattice generated by and is simply the set of all points equivalent to the origin, rotating that lattice by will produce the same lattice as generated by and :

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

Let be the column matrix with entries and :

The Smith decomposition gives , with the unique positive , unimodular matrix:

Hence First[u].mFirst[r]:

But as the first row of is the and the left-hand side of the equation is , the foregoing equation is Bézout's identity. This means the extended GCD is:

Compare to ExtendedGCD:

Use SmithDecomposition to compute the right-extended GCDs of two matrices and :

Let be the matrix formed from the rows of and :

The matrix in the Smith decomposition will be a matrix:

Let and be the top-left and top-right quarters of :

The matrix and thus the matrix are, like , matrices:

Let denote the top half of :

The matrices and have integer entries:

Hence is a right divisor of both and :

Since the determinants of , , and obey , is in fact a right GCD:

Moreover, the first three rows of the Smith decomposition give Bézout's identity :

Hence, the extended right GCD is as follows:

The matrix in the decomposition is a square matrix, where is the number of rows of :

The matrix in the decomposition is a square matrix, where is the number of columns of :

The matrix in the decomposition is a diagonal, rectangular matrix of the same dimensions as :

The matrices and each have determinants of magnitude 1:

The and matrices always have integer values, irrespective of whether the input is integer or rational:

This extends to the real and imaginary parts of the result if the input matrix is complex:

Each diagonal entry in divides its successor:

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 real-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 decomposition can often be found by two iterations of the Hermite decomposition:

The first use of HermiteDecomposition gives and an upper-triangular matrix:

Applying HermiteDecomposition to the Transpose of the triangle matrix gives and :

The middle matrix in the Smith decomposition is the Smith normal form given by SmithReduce: