Hermite and Smith are placeholders for representing the Hermite and Smith Normal Forms respectively. They are used in conjunction with mod as is described below.

Hermite(A, x) mod p computes the Hermite Normal Form (reduced row echelon form) of an m by n rectangular matrix of univariate polynomials in x over the integers modulo p. The polynomial coefficients must be rational or elements of a finite extension field specified by RootOfs. In the case of three arguments, the third argument, U, will be assigned the transformation matrix upon completion, such that Hermite(A) = U &* A.

Smith(A, x) mod p computes the Smith Normal Form of a matrix with univariate polynomial entries in x over the integers modulo p. The coefficients of the polynomial must be either rational or elements of a finite extension field specified by RootOfs. 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.

$A≔\mathrm{Matrix}\left(\left[\left[1+x\,1+x^2\right]\,\left[1+x^2\,1+x^4\right]\right]\right)$

$A≔\left[\begin{array}{cc}1+x& x^2+1\\ x^2+1& x^4+1\end{array}\right]$

$\mathrm{Smith}\left(A\,x\,U\,V\right)\mathbf{mod}2$

$\left[\begin{array}{cc}1+x& 0\\ 0& x^4+x^3+x^2+x\end{array}\right]$

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

$\left[\begin{array}{cc}1& 0\\ x^3+1& 1\end{array}\right]$

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

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

$\mathrm{restart}$

$A≔\mathrm{Matrix}\left(\left[\left[1+x\,1+x^2\right]\,\left[1+x^2\,1+x^4\right]\right]\right)\:$$\mathrm{Hermite}\left(A\,x\,U\right)\mathbf{mod}2$

$\left[\begin{array}{cc}1+x& x^2+1\\ 0& x^4+x^3+x^2+x\end{array}\right]$

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

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

$\mathrm{LinearAlgebra}\left[\mathrm{Transpose}\right]\left(\mathrm{Hermite}\left(\mathrm{LinearAlgebra}\left[\mathrm{Transpose}\right]\left(A\right)\,x\right)\mathbf{mod}2\right)$

$\left[\begin{array}{cc}1+x& 0\\ x^2+1& x^4+x^3+x^2+x\end{array}\right]$