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MatrixPolynomialAlgebra

 PopovForm
 compute the Popov normal form of a Matrix

 Calling Sequence PopovForm(A, x, shifts, out) PopovForm[row](A, x, shifts, out) PopovForm[column](A, x, shifts, out)

Parameters

 A - Matrix x - variable name of the polynomial domain shifts - (optional) equation of the form shifts = obj where obj is a list of one or two lists out - (optional) equation of the form output = obj where obj is one of 'P', 'U', 'rank', 'P_pivots', 'U_pivots' or a list containing one or more of these names; select result objects to compute

Description

 • The PopovForm(A,x) and PopovForm[row](A,x) commands compute the Popov normal form (in row form) of an m x n rectangular Matrix of 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 PopovForm[column](A,x) command computes the Popov normal form (in column form).
 • For row Popov normal form, if m = n and P is nonsingular, then P has the following degree constraints

$\mathrm{deg}\left({P}_{j,i}\right)<\mathrm{deg}\left({P}_{i,i}\right),\mathrm{for all i > j}$

$\mathrm{deg}\left({P}_{j,i}\right)\le \mathrm{deg}\left({P}_{i,i}\right),\mathrm{for all i < j}$

 If $m and P has full row rank, then there is a trailing list of n pivot columns P_pivots such that P[*,P_pivots] is in Popov normal form.
 If $n\le m$ and P has row rank $r, then P has the first $n-r$ rows 0 and there is a trailing list of r columns P_pivots such that P[*,P_pivots] is in Popov normal form. In this case U is a minimal unimodular multiplier and as such there is a list U_pivots of columns such that U[*,U_pivots] is also in Popov normal form.
 The row Popov normal form is obtained by performing elementary row operations on A. This includes interchanging rows, multiplying a row by a unit, and subtracting a polynomial multiple of one row from another. The method used is a fraction-free algorithm by Beckermann, Labahn and Villard. The returned Matrix objects have the property that $P=U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A$ for row Popov normal form.
 • The output option (out) determines the content of the returned expression sequence.
 As determined by the out option, an expression sequence containing one or more of the factors P (the Popov normal form), U (the unimodular transformation Matrix), rank (the rank of the matrix), P_pivots, or U_pivots (the pivot columns of P and U, respectively -- interesting in the non-full rank case) is returned. If obj is a list, the objects are returned in the order specified in the list.
 • The shifts option is an optional input that allows the user to shift the degree constraints on both the Popov form and the minimal multiplier (in the non-full row rank case).
 • The row Popov normal form and the column Popov normal form is related in the following way.  If P is the row Popov normal form of A, P^T is the column Popov normal form of A^T.

Examples

 > $\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right):$
 > $A≔⟨⟨{z}^{3}-{z}^{2},{z}^{3}-2{z}^{2}+2z-2⟩|⟨{z}^{3}-2{z}^{2}-1,{z}^{3}-3{z}^{2}+3z-4⟩⟩$
 ${A}{:=}\left[\begin{array}{cc}{{z}}^{{3}}{-}{{z}}^{{2}}& {{z}}^{{3}}{-}{2}{}{{z}}^{{2}}{-}{1}\\ {{z}}^{{3}}{-}{2}{}{{z}}^{{2}}{+}{2}{}{z}{-}{2}& {{z}}^{{3}}{-}{3}{}{{z}}^{{2}}{+}{3}{}{z}{-}{4}\end{array}\right]$ (1)
 > $P≔\mathrm{PopovForm}[\mathrm{row}]\left(A,z\right)$
 ${P}{:=}\left[\begin{array}{cc}{{z}}^{{2}}{-}{z}{+}{1}& {0}\\ \frac{{1}}{{2}}& {1}\end{array}\right]$ (2)
 > $P,U≔\mathrm{PopovForm}[\mathrm{row}]\left(A,z,\mathrm{output}=\left['P','U'\right]\right)$
 ${P}{,}{U}{:=}\left[\begin{array}{cc}{{z}}^{{2}}{-}{z}{+}{1}& {0}\\ \frac{{1}}{{2}}& {1}\end{array}\right]{,}\left[\begin{array}{cc}{-}\frac{{1}}{{2}}{}{{z}}^{{3}}{+}\frac{{3}}{{2}}{}{{z}}^{{2}}{-}\frac{{3}}{{2}}{}{z}{+}{2}& \frac{{1}}{{2}}{}{{z}}^{{3}}{-}{{z}}^{{2}}{-}\frac{{1}}{{2}}\\ \frac{{1}}{{4}}{}{z}& {-}\frac{{1}}{{4}}{}{z}{-}\frac{{1}}{{4}}\end{array}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{expand},P-U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\right)$
 $\left[\begin{array}{rr}{0}& {0}\\ {0}& {0}\end{array}\right]$ (4)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(U\right)$
 ${-}\frac{{1}}{{2}}$ (5)
 > $P≔\mathrm{PopovForm}[\mathrm{column}]\left(A,z\right)$
 ${P}{:=}\left[\begin{array}{cc}{z}& {-}{1}\\ {1}& {-}{1}{+}{z}\end{array}\right]$ (6)
 > $P,U≔\mathrm{PopovForm}[\mathrm{column}]\left(A,z,\mathrm{output}=\left['P','U'\right]\right)$
 ${P}{,}{U}{:=}\left[\begin{array}{cc}{z}& {-}{1}\\ {1}& {-}{1}{+}{z}\end{array}\right]{,}\left[\begin{array}{cc}{-}\frac{{1}}{{2}}{}{{z}}^{{2}}{+}\frac{{3}}{{2}}{}{z}{-}\frac{{1}}{{2}}& \frac{{1}}{{2}}{}{{z}}^{{2}}{-}\frac{{1}}{{2}}{}{z}{-}\frac{{3}}{{2}}\\ \frac{{1}}{{2}}{}{{z}}^{{2}}{-}{z}& {-}\frac{{1}}{{2}}{}{{z}}^{{2}}{+}{1}\end{array}\right]$ (7)
 > $\mathrm{map}\left(\mathrm{expand},P-A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}U\right)$
 $\left[\begin{array}{rr}{0}& {0}\\ {0}& {0}\end{array}\right]$ (8)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(U\right)$
 ${-}\frac{{1}}{{2}}$ (9)

Low rank matrix:

 > $A≔⟨⟨3z-6,-3z+3,2z+3,z⟩|⟨-3z,3z,-2,-1⟩|⟨6,-3,-2z-1,-z+1⟩⟩$
 ${A}{:=}\left[\begin{array}{ccc}{3}{}{z}{-}{6}& {-}{3}{}{z}& {6}\\ {-}{3}{}{z}{+}{3}& {3}{}{z}& {-}{3}\\ {2}{}{z}{+}{3}& {-}{2}& {-}{2}{}{z}{-}{1}\\ {z}& {-}{1}& {-}{z}{+}{1}\end{array}\right]$ (10)
 > $P,U,r≔\mathrm{PopovForm}[\mathrm{row}]\left(A,z,\mathrm{output}=\left['P','U','\mathrm{rank}'\right]\right)$
 ${P}{,}{U}{,}{r}{:=}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {-}{1}& {1}& {0}\\ {-}{1}& {0}& {1}\end{array}\right]{,}\left[\begin{array}{cccc}{3}{}{{z}}^{{2}}{+}\frac{{3}}{{2}}{}{z}{+}{3}& {3}{}{{z}}^{{2}}{+}\frac{{3}}{{2}}{}{z}{+}{6}& \frac{{1}}{{2}}{}{z}& {0}\\ {-}\frac{{9}}{{2}}& {-}\frac{{9}}{{2}}& {-}\frac{{1}}{{2}}& {1}\\ {-}\frac{{3}}{{2}}{-}{3}{}{z}& {-}\frac{{3}}{{2}}{-}{3}{}{z}& {-}\frac{{1}}{{2}}& {0}\\ {3}& {3}& {0}& {0}\end{array}\right]{,}{2}$ (11)
 > $P,U,r,\mathrm{II},K≔\mathrm{PopovForm}[\mathrm{row}]\left(A,z,\mathrm{output}=\left['P','U','\mathrm{rank}','\mathrm{P_pivots}','\mathrm{U_pivots}'\right]\right)$
 ${P}{,}{U}{,}{r}{,}{\mathrm{II}}{,}{K}{:=}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {-}{1}& {1}& {0}\\ {-}{1}& {0}& {1}\end{array}\right]{,}\left[\begin{array}{cccc}{3}{}{{z}}^{{2}}{+}\frac{{3}}{{2}}{}{z}{+}{3}& {3}{}{{z}}^{{2}}{+}\frac{{3}}{{2}}{}{z}{+}{6}& \frac{{1}}{{2}}{}{z}& {0}\\ {-}\frac{{9}}{{2}}& {-}\frac{{9}}{{2}}& {-}\frac{{1}}{{2}}& {1}\\ {-}\frac{{3}}{{2}}{-}{3}{}{z}& {-}\frac{{3}}{{2}}{-}{3}{}{z}& {-}\frac{{1}}{{2}}& {0}\\ {3}& {3}& {0}& {0}\end{array}\right]{,}{2}{,}\left[{2}{,}{3}\right]{,}\left[{2}{,}{4}\right]$ (12)
 > $P,U,r≔\mathrm{PopovForm}[\mathrm{column}]\left(A,z,\mathrm{output}=\left['P','U','\mathrm{rank}'\right]\right)$
 ${P}{,}{U}{,}{r}{:=}\left[\begin{array}{ccc}{0}& {-}{z}& {-}{6}\\ {0}& {z}& {3}\\ {0}& {-}\frac{{2}}{{3}}& {1}{+}{2}{}{z}\\ {0}& {-}\frac{{1}}{{3}}& {-}{1}{+}{z}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& {0}& {1}\\ {1}& \frac{{1}}{{3}}& {1}\\ {1}& {0}& {0}\end{array}\right]{,}{2}$ (13)
 > $P,U,r,\mathrm{II},K≔\mathrm{PopovForm}[\mathrm{column}]\left(A,z,\mathrm{output}=\left['P','U','\mathrm{rank}','\mathrm{P_pivots}','\mathrm{U_pivots}'\right]\right)$
 ${P}{,}{U}{,}{r}{,}{\mathrm{II}}{,}{K}{:=}\left[\begin{array}{ccc}{0}& {-}{z}& {-}{6}\\ {0}& {z}& {3}\\ {0}& {-}\frac{{2}}{{3}}& {1}{+}{2}{}{z}\\ {0}& {-}\frac{{1}}{{3}}& {-}{1}{+}{z}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& {0}& {1}\\ {1}& \frac{{1}}{{3}}& {1}\\ {1}& {0}& {0}\end{array}\right]{,}{2}{,}\left[{2}{,}{4}\right]{,}\left[{3}\right]$ (14)

[1,-2,0]-shifted Popov form:

 > $P,U,r,\mathrm{II},K≔\mathrm{PopovForm}[\mathrm{row}]\left(A,z,\mathrm{shifts}=\left[\left[1,-2,0\right]\right],\mathrm{output}=\left['P','U','\mathrm{rank}','\mathrm{P_pivots}','\mathrm{U_pivots}'\right]\right)$
 ${P}{,}{U}{,}{r}{,}{\mathrm{II}}{,}{K}{:=}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {-}{1}& {1}& {0}\\ {-}{1}& {0}& {1}\end{array}\right]{,}\left[\begin{array}{cccc}{3}{}{{z}}^{{2}}{+}\frac{{3}}{{2}}{}{z}{+}{3}& {3}{}{{z}}^{{2}}{+}\frac{{3}}{{2}}{}{z}{+}{6}& \frac{{1}}{{2}}{}{z}& {0}\\ {-}\frac{{9}}{{2}}& {-}\frac{{9}}{{2}}& {-}\frac{{1}}{{2}}& {1}\\ {-}\frac{{3}}{{2}}{-}{3}{}{z}& {-}\frac{{3}}{{2}}{-}{3}{}{z}& {-}\frac{{1}}{{2}}& {0}\\ {3}& {3}& {0}& {0}\end{array}\right]{,}{2}{,}\left[{2}{,}{3}\right]{,}\left[{2}{,}{4}\right]$ (15)

[2,2,0,0]-shifted Popov form:

 > $P,U,r,\mathrm{II},K≔\mathrm{PopovForm}[\mathrm{column}]\left(A,z,\mathrm{shifts}=\left[\left[2,2,0,0\right]\right],\mathrm{output}=\left['P','U','\mathrm{rank}','\mathrm{P_pivots}','\mathrm{U_pivots}'\right]\right)$
 ${P}{,}{U}{,}{r}{,}{\mathrm{II}}{,}{K}{:=}\left[\begin{array}{ccc}{0}& {-}{{z}}^{{2}}{+}{z}{-}{2}& {2}{}{{z}}^{{2}}{+}{z}{+}{4}\\ {0}& {{z}}^{{2}}{-}{z}{+}{1}& {-}{2}{}{{z}}^{{2}}{-}{z}{-}{2}\\ {0}& {1}& {0}\\ {0}& {0}& {1}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& \frac{{1}}{{3}}& {-}\frac{{2}}{{3}}\\ {1}& \frac{{1}}{{3}}{}{z}& {-}{1}{-}\frac{{2}}{{3}}{}{z}\\ {1}& {0}& {0}\end{array}\right]{,}{2}{,}\left[{3}{,}{4}\right]{,}\left[{3}\right]$ (16)

Popov form with [0,-3]-shift for unimodular multiplier:

 > $A≔⟨⟨-{z}^{3}+4{z}^{2}+z+1,-{z}^{2}+7z+4⟩|⟨z-1,z+2⟩|⟨2{z}^{2}+2z-2,{z}^{2}+6z+6⟩|⟨-{z}^{2},-2z⟩⟩$
 ${A}{:=}\left[\begin{array}{cccc}{-}{{z}}^{{3}}{+}{4}{}{{z}}^{{2}}{+}{z}{+}{1}& {-}{1}{+}{z}& {2}{}{{z}}^{{2}}{+}{2}{}{z}{-}{2}& {-}{{z}}^{{2}}\\ {-}{{z}}^{{2}}{+}{7}{}{z}{+}{4}& {z}{+}{2}& {{z}}^{{2}}{+}{6}{}{z}{+}{6}& {-}{2}{}{z}\end{array}\right]$ (17)
 > $P,U,r,\mathrm{II},K≔\mathrm{PopovForm}[\mathrm{row}]\left(A,z,\mathrm{shifts}=\left[\left[0,0,0,0\right],\left[0,-3\right]\right],\mathrm{output}=\left['P','U','\mathrm{rank}','\mathrm{P_pivots}','\mathrm{U_pivots}'\right]\right)$
 ${P}{,}{U}{,}{r}{,}{\mathrm{II}}{,}{K}{:=}\left[\begin{array}{cccc}{{z}}^{{3}}{-}{6}{}{{z}}^{{2}}{+}{13}{}{z}{+}{7}& {z}{+}{5}& {14}{+}{10}{}{z}& {{z}}^{{2}}{-}{4}{}{z}\\ {-}{{z}}^{{2}}{+}{7}{}{z}{+}{4}& {z}{+}{2}& {{z}}^{{2}}{+}{6}{}{z}{+}{6}& {-}{2}{}{z}\end{array}\right]{,}\left[\begin{array}{rr}{-}{1}& {2}\\ {0}& {1}\end{array}\right]{,}{2}{,}\left[{1}{,}{3}\right]{,}\left[{}\right]$ (18)
 > $\mathrm{map}\left(\mathrm{expand},P-U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\right)$
 $\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (19)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(U\right)$
 ${-}{1}$ (20)

Popov form with [0,-3,0,0]-shift for unimodular multiplier:

 > $P,U,r,\mathrm{II},K≔\mathrm{PopovForm}[\mathrm{column}]\left(A,z,\mathrm{shifts}=\left[\left[0,0\right],\left[0,-3,0,0\right]\right],\mathrm{output}=\left['P','U','\mathrm{rank}','\mathrm{P_pivots}','\mathrm{U_pivots}'\right]\right)$
 ${P}{,}{U}{,}{r}{,}{\mathrm{II}}{,}{K}{:=}\left[\begin{array}{cccc}{0}& {0}& {z}& {-}{1}\\ {0}& {0}& {2}& {z}\end{array}\right]{,}\left[\begin{array}{cccc}{-}\frac{{2}}{{21}}{}{z}{+}\frac{{1}}{{7}}& {-}{{z}}^{{2}}{-}{2}{}{z}& \frac{{2}}{{7}}{+}\frac{{1}}{{7}}{}{z}& {-}\frac{{1}}{{21}}{}{z}{-}\frac{{3}}{{7}}\\ {1}& {0}& {0}& {0}\\ {-}\frac{{3}}{{7}}{+}\frac{{2}}{{21}}{}{z}& {{z}}^{{2}}{-}{z}& {-}\frac{{1}}{{7}}{}{z}{+}\frac{{1}}{{7}}& \frac{{1}}{{21}}{}{z}{+}\frac{{2}}{{7}}\\ \frac{{2}}{{21}}{}{{z}}^{{2}}{-}\frac{{1}}{{3}}{}{z}{-}\frac{{4}}{{21}}& {{z}}^{{3}}{-}{9}{}{z}{-}{7}& {-}\frac{{1}}{{7}}{}{{z}}^{{2}}{+}\frac{{9}}{{7}}& \frac{{1}}{{21}}{}{{z}}^{{2}}{+}\frac{{1}}{{3}}{}{z}{-}\frac{{23}}{{21}}\end{array}\right]{,}{2}{,}\left[{1}{,}{2}\right]{,}\left[{2}{,}{4}\right]$ (21)
 > $\mathrm{map}\left(\mathrm{expand},P-A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}U\right)$
 $\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (22)

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.
 Beckermann, B.; Labahn, G.; and Villard, G. "Shifted Normal Forms of General Polynomial Matrices." University of Waterloo, Technical Report, Department of Computer Science, (2001).
 Beckermann, B.; Labahn, G.; and Villard, G. "Shifted Normal Forms of Polynomial Matrices" ISSAC'99 (1999): 189-196.