steventhornton / WeyrForm

A Maple function for computing the Weyr canonical form

Weyr Canonical Form

This repository contains a Maple function for computing the Weyr canonical form of a square matrix.

The function WeyrForm expects a square matrix as input. It will return the Weyr form of the input by default. The optional output argument can be used to specify what should be returned:

• output = W will return the Weyr form
• output = Q will return an invertible matrix Q such that MatrixInverse(Q).A.Q = W
• output = [W, Q] will return both the Weyr form and invertible matrix Q
• output = [Q, W] will return the invertible matrix followed by the Weyr form

The Example.mw Maple worksheet includes 4 examples.

A poster explaining how this implementation works can be found here.

Example

read("WeyrForm.mpl");
with(LinearAlgebra):

A := Matrix([[-1,  0,  0,  2,  1,  0,  0],
             [-1,  0,  0,  1,  1,  0,  0],
             [ 6, -2, -2,  4,  2, -2,  4],
             [-2,  1,  1,  0,  0,  1, -1],
             [ 3, -1, -1,  2,  1, -1,  2],
             [-5,  2,  2, -5, -3,  2, -4],
             [ 2,  0,  0, -4, -2,  0,  0]]);

Q, W := WeyrForm(A, output = [Q, W]);

Relevant References

• O'Meara, K. et al (2011). Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form. Oxford University Press, USA.
• Helene Shapiro. The Weyr Characteristic. The American Mathematical Monthly, 106(10):919–929, 1999.
• Eduard Weyr. Répartition des matrices en espèces et formation de toutes les espèces. CR Acad. Sci. Paris, 100:966–969, 1885.

License

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.