Important: The linalg package has been deprecated. Use the superseding packages, LinearAlgebra and VectorCalculus, instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The function ismith computes the Smith normal form S of an n by m rectangular matrix of integers.

If two n by n matrices have the same Smith normal form, they are equivalent.

The Smith normal form is a diagonal matrix where

Hence if n = m and rank(A) = n then .

The Smith normal form is obtained by doing elementary row and column operations.  This includes interchanging rows (columns), multiplying through a row (column) by -1, and adding integral multiples of one row (column) to another.

Although the rank and determinant can be easily obtained from S this is not an efficient method for computing these quantities except that this may yield a partial factorization of  without doing any explicit factorizations.

In the case of three arguments, the second argument U and the third argument V will be assigned the transformation matrices on output, such that smith(A) = U &* A &* V.

The command with(linalg,ismith) allows the use of the abbreviated form of this command.