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 $\left|\det \left(A\right)\right|=\prod _{i=1}^n{S}_{i,i}$.

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 $\det \left(A\right)$ 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.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$H≔\mathrm{array}\left(\left[\left[9\,-36\,30\right]\,\left[-36\,192\,-180\right]\,\left[30\,-180\,180\right]\right]\right)$

$H≔\left[\begin{array}{ccc}9& \mathrm{-36}& 30\\ \mathrm{-36}& 192& \mathrm{-180}\\ 30& \mathrm{-180}& 180\end{array}\right]$

$\mathrm{ismith}\left(H\right)$

$\left[\begin{array}{ccc}3& 0& 0\\ 0& 12& 0\\ 0& 0& 60\end{array}\right]$

$A≔\mathrm{array}\left(\left[\left[13\,5\,7\right]\,\left[17\,31\,39\right]\right]\right)$

$A≔\left[\begin{array}{ccc}13& 5& 7\\ 17& 31& 39\end{array}\right]$

$B≔\mathrm{ismith}\left(A\,U\,V\right)$

$B≔\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\end{array}\right]$

$\mathrm{eval}\left(U\right)$

$\left[\begin{array}{cc}0& 1\\ \mathrm{-1}& 1\end{array}\right]$

$\mathrm{eval}\left(V\right)$

$\left[\begin{array}{ccc}\mathrm{-4}& 7& \mathrm{-11}\\ \mathrm{-72}& 127& \mathrm{-194}\\ 59& \mathrm{-104}& 159\end{array}\right]$

$\mathrm{evalm}\left(\left(U\&*A\right)\&*V-B\right)$

$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]$