Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[SmithForm], instead.

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

The Smith normal form of a matrix with univariate polynomial entries in x over a field F is computed. Thus the polynomials are then regarded as elements of the Euclidean domain F[x].

This routine is only as powerful as Maple's normal function, since at present it only understands the field Q of rational numbers and rational functions over Q.

The Smith normal form of a matrix is a diagonal matrix S obtained by doing elementary row and column operations.  The diagonal entries satisfy the following property for all $n\le \mathrm{rank}\left(A\right)$: $\prod _{i=1}^n{S}_{i,i}$ is equal to the (monic) greatest common divisor of all n by n minors of A.

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

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

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

$A≔\mathrm{matrix}\left(\left[\left[1-x\,y-xy\right]\,\left[0\,1-x^2\right]\right]\right)$

$A≔\left[\begin{array}{cc}1-x& -xy+y\\ 0& -x^2+1\end{array}\right]$

$\mathrm{smith}\left(A\,x\right)$

$\left[\begin{array}{cc}-1+x& 0\\ 0& x^2-1\end{array}\right]$

$H≔\mathrm{inverse}\left(\mathrm{hilbert}\left(2\,x\right)\right)$

$H≔\left[\begin{array}{cc}-{\left(-3+x\right)}^2\left(-2+x\right)& \left(-3+x\right)\left(-2+x\right)\left(-4+x\right)\\ \left(-3+x\right)\left(-2+x\right)\left(-4+x\right)& -{\left(-3+x\right)}^2\left(-4+x\right)\end{array}\right]$

$\mathrm{smith}\left(H\,x\right)$

$\left[\begin{array}{cc}-3+x& 0\\ 0& x^3-9x^2+26x-24\end{array}\right]$

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

$B≔\left[\begin{array}{cc}-1+x& 0\\ 0& x^2-1\end{array}\right]$

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

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

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

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

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

$\left[\begin{array}{cc}\left(-1+x\right)\left(1-y\right)+xy-y+1-x& -\left(-1+x\right)y+xy-y\\ \left(x+1\right)\left(1-x\right)\left(1-y\right)+\left(x+1\right)\left(-xy+y\right)+x^2-1& -\left(x+1\right)\left(1-x\right)y+\left(x+1\right)\left(-xy+y\right)\end{array}\right]$