The MatrixLCLM(A, B, x) command computes a least common left multiple of two matrices of polynomials. Both input matrices of polynomials can be square or rectangular but must have the same number of columns. The entries are either univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q, that is, univariate polynomials in x with coefficients in Q(a1,...,an).

The matrix of polynomials $\[A^T{B}^T\]\^T$ must have more rows than columns and full column rank.

The MatrixLCRM(A, B, x) command computes a least common right multiple of two matrices of polynomials.  The matrix of polynomials $\[AB\]$ must have more columns than rows and full row rank.

The method is a fraction-free algorithm by Beckermann and Labahn that computes a matrix GCD using Mahler systems.

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

$A≔\mathrm{Matrix}\left(2\,2\,\left[\left[-9z^2-3z+1\,12z^2+10z\right]\,\left[-3z^3+2z^2-z\,4z^3+2z-2z^2\right]\right]\right)\:$

$B≔\mathrm{Matrix}\left(2\,2\,\left[\left[-3z^3+6z^2+5z+1\,-12z^2-13z\right]\,\left[z^4+z^3+z^2\,-4z^3-3z+3z^2\right]\right]\right)\:$

$C≔\mathrm{MatrixLCLM}\left(A\,B\,z\right)$

$C≔\left[\begin{array}{cc}108z^5-228z^4-92z^3+66z^2+27z+2& -144z^5+504z^4+28z^3-128z^2-26z\\ 36z^5-84z^4-24z^3+26z^2+7z& -48z^5+184z^4-12z^3-40z^2-6z\end{array}\right]$

$C≔\mathrm{MatrixLCRM}\left(A\,B\,z\right)$

$C≔\left[\begin{array}{cc}60z^4+59z^3+12z^2+10z+2& 12z^3+z^2-13z\\ 20z^5-13z^4+17z^3+2z^2& 4z^4-7z^3+6z^2-3z\end{array}\right]$

Beckermann, B., and Labahn, G. "Fraction-free Computation of Matrix Rational Interpolants and Matrix GCDs." SIAM Journal on Matrix Analysis and Applications, Vol. 22 No.1, (2000): 114-144.