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MatrixPolynomialAlgebra

 SmithForm
 reduce a Matrix to Smith normal form

 Calling Sequence SmithForm(A, x, out)

Parameters

 A - Matrix x - (optional) variable out - (optional) equation of the form output = obj where obj is one of 'S', 'U', or 'V', or a list containing one or more of these names; select result objects to compute

Description

 • The SmithForm(A) command returns the Smith normal form S of a Matrix A with univariate polynomial entries in x over a field F. Thus, the polynomials are regarded as elements of the Euclidean domain F[x].
 The Smith normal form of a Matrix is a diagonal Matrix S obtained by doing elementary row and column operations. The diagonal entries satisfy the property that for all n <= Rank(A), product(S[i, i], i=1..n) is equal to the (monic) greatest common divisor of all n x n (determinant) minors of A.
 • 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 S (the Smith normal form), U (the left-reducing Matrix ), or V (the right-reducing Matrix) is returned. If obj is a list, the objects are returned in the order specified in the list.
 The returned Matrix objects have the property that S = U . A . V.
 • Note: The MatrixPolynomialAlgegra:-SmithForm command calls the LinearAlgebra:-SmithForm routine.

Examples

 > $\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,2x,2{x}^{2}+2x\right],\left[1,6x,6{x}^{2}+6x\right],\left[1,3,x\right]\right]\right)$
 ${A}{:=}\left[\begin{array}{ccc}{1}& {2}{}{x}& {2}{}{{x}}^{{2}}{+}{2}{}{x}\\ {1}& {6}{}{x}& {6}{}{{x}}^{{2}}{+}{6}{}{x}\\ {1}& {3}& {x}\end{array}\right]$ (1)
 > $S≔\mathrm{SmithForm}\left(A\right)$
 ${S}{:=}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {{x}}^{{2}}{+}\frac{{3}}{{2}}{}{x}\end{array}\right]$ (2)
 > $\mathrm{LinearAlgebra}[\mathrm{Determinant}]\left(A\right)$
 ${-}{8}{}{{x}}^{{2}}{-}{12}{}{x}$ (3)
 > $\frac{}{\mathrm{lcoeff}\left(\right)}$
 ${{x}}^{{2}}{+}\frac{{3}}{{2}}{}{x}$ (4)
 > $U,V≔\mathrm{SmithForm}\left(A,x,\mathrm{output}=\left['U','V'\right]\right):$
 > $\mathrm{map}\left(\mathrm{expand},U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}V\right)$
 $\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {{x}}^{{2}}{+}\frac{{3}}{{2}}{}{x}\end{array}\right]$ (5)