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MatrixPolynomialAlgebra

 MatrixLCLM
 compute a least-common left multiple of 2 matrices of polynomials
 MatrixLCRM
 compute a least-common right multiple of 2 matrices of polynomials

 Calling Sequence MatrixLCLM(A, B, x) MatrixLCRM(A, B, x)

Parameters

 A - Matrix of polynomials B - Matrix of polynomials x - variable name of the polynomial domain

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right):$
 > $A≔\mathrm{Matrix}\left(2,2,\left[\left[-9{z}^{2}-3z+1,12{z}^{2}+10z\right],\left[-3{z}^{3}+2{z}^{2}-z,4{z}^{3}+2z-2{z}^{2}\right]\right]\right):$
 > $B≔\mathrm{Matrix}\left(2,2,\left[\left[-3{z}^{3}+6{z}^{2}+5z+1,-12{z}^{2}-13z\right],\left[{z}^{4}+{z}^{3}+{z}^{2},-4{z}^{3}-3z+3{z}^{2}\right]\right]\right):$

Left matrix LCMs:

 > $C≔\mathrm{MatrixLCLM}\left(A,B,z\right)$
 ${C}{:=}\left[\begin{array}{cc}{108}{}{{z}}^{{5}}{-}{228}{}{{z}}^{{4}}{-}{92}{}{{z}}^{{3}}{+}{66}{}{{z}}^{{2}}{+}{27}{}{z}{+}{2}& {-}{144}{}{{z}}^{{5}}{+}{504}{}{{z}}^{{4}}{+}{28}{}{{z}}^{{3}}{-}{128}{}{{z}}^{{2}}{-}{26}{}{z}\\ {36}{}{{z}}^{{5}}{-}{84}{}{{z}}^{{4}}{-}{24}{}{{z}}^{{3}}{+}{26}{}{{z}}^{{2}}{+}{7}{}{z}& {-}{48}{}{{z}}^{{5}}{+}{184}{}{{z}}^{{4}}{-}{12}{}{{z}}^{{3}}{-}{40}{}{{z}}^{{2}}{-}{6}{}{z}\end{array}\right]$ (1)

Right matrix LCMs:

 > $C≔\mathrm{MatrixLCRM}\left(A,B,z\right)$
 ${C}{:=}\left[\begin{array}{cc}{60}{}{{z}}^{{4}}{+}{59}{}{{z}}^{{3}}{+}{12}{}{{z}}^{{2}}{+}{10}{}{z}{+}{2}& {12}{}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{13}{}{z}\\ {20}{}{{z}}^{{5}}{-}{13}{}{{z}}^{{4}}{+}{17}{}{{z}}^{{3}}{+}{2}{}{{z}}^{{2}}& {4}{}{{z}}^{{4}}{-}{7}{}{{z}}^{{3}}{+}{6}{}{{z}}^{{2}}{-}{3}{}{z}\end{array}\right]$ (2)

References

 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.