 LeftDivision - Maple Help

MatrixPolynomialAlgebra

 LeftDivision
 compute a left quotient and remainder of 2 matrices of polynomials
 RightDivision
 compute a right quotient and remainder of 2 matrices of polynomials Calling Sequence LeftDivision(A, B, x) RightDivision(A, B, x) Parameters

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

 • The LeftDivision(A, B, x) command computes a left quotient Q and a remainder R such that $A=B·Q+R$ where ${B}^{\left(-1\right)}·R$ is strictly proper.  That is,  $\underset{z\to \mathrm{\infty }}{lim}{B}^{\left(-1\right)}\left(z\right).R\left(z\right)$ is a zero matrix. The input matrices must have the same number of rows, and B must be a square nonsingular matrix of polynomials.
 • The RightDivision(A, B, x) command computes a right quotient Q and a remainder R such that $A=Q·B+R$ where $R·{B}^{\left(-1\right)}$ is strictly proper.  That is,  $\underset{z\to \mathrm{\infty }}{lim}R\left(z\right).{B}^{\left(-1\right)}\left(z\right)$ is a zero matrix. The input matrices must have the same number of columns, and B must be a square nonsingular matrix of polynomials.
 • The quotient $Q$ and the remainder $R$ are returned in a list. Examples

 > $\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right):$
 > $A≔\mathrm{Matrix}\left(2,2,\left[\left[-9{z}^{2}-3z+1,12{z}^{2}+10z\right],\left[-3{z}^{3}+2{z}^{2}-z,4{z}^{3}+2z-2{z}^{2}\right]\right]\right):$
 > $B≔\mathrm{Matrix}\left(2,2,\left[\left[-3{z}^{3}+6{z}^{2}+5z+1,-12{z}^{2}-13z\right],\left[{z}^{4}+{z}^{3}+{z}^{2},-4{z}^{3}-3z+3{z}^{2}\right]\right]\right):$
 > $Q,R≔\mathrm{op}\left(\mathrm{LeftDivision}\left(A,B,z\right)\right)$
 ${Q}{,}{R}{≔}\left[\begin{array}{cc}{0}& {0}\\ \frac{{3}}{{4}}& {-1}\end{array}\right]{,}\left[\begin{array}{cc}\frac{{27}{}{z}}{{4}}{+}{1}& {-}{3}{}{z}\\ {-}\frac{{1}}{{4}}{}{{z}}^{{2}}{+}\frac{{5}}{{4}}{}{z}& {{z}}^{{2}}{-}{z}\end{array}\right]$ (1)
 > $\mathrm{map}\left(\mathrm{expand},A-\left(B·Q+R\right)\right)$
 $\left[\begin{array}{cc}{0}& {0}\\ {0}& {0}\end{array}\right]$ (2)
 > $\mathrm{map}\left(f↦\mathrm{limit}\left(f,z=\mathrm{\infty }\right),\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(B\right)·R\right)$
 $\left[\begin{array}{cc}{0}& {0}\\ {0}& {0}\end{array}\right]$ (3)
 > $Q,R≔\mathrm{op}\left(\mathrm{RightDivision}\left(A,B,z\right)\right)$
 ${Q}{,}{R}{≔}\left[\begin{array}{cc}{-}\frac{{1}}{{2}}& {0}\\ {-}\frac{{z}}{{6}}{+}\frac{{13}}{{36}}& {-}\frac{{1}}{{2}}\end{array}\right]{,}\left[\begin{array}{cc}{-}{6}{}{{z}}^{{2}}{-}\frac{{1}}{{2}}{}{z}{+}\frac{{3}}{{2}}{-}\frac{{3}}{{2}}{}{{z}}^{{3}}& {6}{}{{z}}^{{2}}{+}\frac{{7}}{{2}}{}{z}\\ {-}\frac{{5}}{{12}}{}{{z}}^{{3}}{+}\frac{{7}}{{6}}{}{{z}}^{{2}}{-}\frac{{95}}{{36}}{}{z}{-}\frac{{13}}{{36}}& \frac{{5}}{{3}}{}{{z}}^{{2}}{+}\frac{{187}}{{36}}{}{z}\end{array}\right]$ (4)
 > $\mathrm{map}\left(\mathrm{expand},A-\left(Q·B+R\right)\right)$
 $\left[\begin{array}{cc}{0}& {0}\\ {0}& {0}\end{array}\right]$ (5)
 > $\mathrm{map}\left(f↦\mathrm{limit}\left(f,z=\mathrm{\infty }\right),R·\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(B\right)\right)$
 $\left[\begin{array}{cc}{0}& {0}\\ {0}& {0}\end{array}\right]$ (6)