compute a column-reduced form of a Matrix
compute a row-reduced form of a Matrix
ColumnReducedForm(A, x, U)
RowReducedForm(A, x, U)
variable name of the polynomial domain
(optional) name to return unimodular multiplier
The ColumnReducedForm(A,x) command computes a column-reduced 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 RowReducedForm(A,x) command computes a row-reduced form over such domains.
A column-reduced form is one in which the column leading coefficient matrix has the same column rank as the rank of the matrix of polynomials. A row reduced form has the same properties but with respect to the leading row.
The column-reduced form is obtained by elementary column operations, which include interchanging columns, multiplying a column by a unit, or subtracting a polynomial multiple of one column from another. The row-reduced form uses similar row operations. The method used is a fraction-free algorithm by Beckermann and Labahn.
The optional third argument returns a unimodular matrix of elementary operations having the property that P=A.U in the column-reduced case and P=U.A in the row-reduced case.
A ≔ z3−z2,z3−2⁢z2+2⁢z−2|z3−2⁢z2−1,z3−3⁢z2+3⁢z−4
A ≔ z3−z2z3−2⁢z2−1z3−2⁢z2+2⁢z−2z3−3⁢z2+3⁢z−4
P ≔ ColumnReducedForm⁡A,z
P ≔ z1+3⁢z14−z
d ≔ Degree[column]⁡P,z
d ≔ 1,1
C ≔ Coeff[column]⁡P,z,d
C ≔ 130−1
P ≔ ColumnReducedForm⁡A,z,'U'
P ≔ RowReducedForm⁡A,z
P ≔ z23⁢z2−z+112
d ≔ Degree[row]⁡P,z
d ≔ 2,0
C ≔ Coeff[row]⁡P,z,d
C ≔ 1312
P ≔ RowReducedForm⁡A,z,'U'
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.
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