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ComprehensiveFrobeniusForm.md

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ComprehensiveFrobeniusForm

The ComprehensiveFrobeniusForm function will compute a complete case discussion of the Frobenius (rational) normal form of a matrix of multivariate polynomials.

Calling Sequence

ComprehensiveFrobeniusForm(A, R, options)
ComprehensiveFrobeniusForm(A, rs, R, options)
ComprehensiveFrobeniusForm(A, cs, R, options)
ComprehensiveFrobeniusForm(A, F, R, options)
ComprehensiveFrobeniusForm(A, F, H, R, options)

Input

Variable Description
A Square matrix of multivariate polynomials over R
rs Regular system
cs Constructible set
F List of polynomials over R representing equality constraints on the parameters
H List of polynomials over R representing inequation constraints on the parameters
R Polynomial ring

Options

Option name Default value Description
'outputType' 'CS' When set to 'CS' or 'ConstructibleSet' the output will contain constructible sets, when set to 'RS' or 'RegularSystem', the output will contain regular systems.
'outputMatrices' 'F' Which matrices to return, value can be either 'F' or 'Q' or a list containing any combination of 'F', and 'Q'
'lazy' false When true, this method will only return one branch of the computation. This option is not currently implemented.
'algorithm' 'snf_minors' Specify which algorithm to use, currently the 'snf_minors' algorithm is the only one implemented.

Option Compatibility

  • Only the Frobenius form can be returned when the snf_minors algorithm is used. The transformation matrix can not be computed with this algorithm.

Output

Output structure Options
[F, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = 'F'
[F, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = 'F'
[Q, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = 'Q'
[Q, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = 'Q'
[F, Q, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = ['F', 'Q']
[F, Q, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = ['F', 'Q']

Where F is the Frobenius form of the input matrix for all parameter values in the zero set of cs or rs. Q is the similarity transformation matrix such that F = Q^(-1) A Q. Together, all the constructible sets or regular systems form a partition of the input constructible set, regular system or set of equations and inequations.

Example

with(RegularChains):
with(ParametricMatrixTools):

# Set up the matrix
R := PolynomialRing([a]):
A := Matrix([[ 0,  a,  1,  1,  1],
             [ 2, -2,  0, -2, -4],
             [ 0,  0,  1,  1,  3],
             [-6,  0, -3, -1, -3],
             [ 2,  2,  2,  2,  4]]):

# Compute the Frobenius form of A
result := ComprehensiveFrobeniusForm(A, [], R)

# Print the result
result[1][1], Display(result[1][2], R);
    [0, 0, 0, 0, -8a + 16]
    [1, 0, 0, 0,  4a -  8]          {a-4 <> 0
    [0, 1, 0, 0, -4a + 12]          {a-1 <> 0
    [0, 0, 1, 0,  2a -  6]          {23*a^2-61*a+68 <> 0
    [0, 0, 0, 1,        2]

result[2][1], Display(result[2][2], R);
    [0, 0, 0, 0, -8a + 16]
    [1, 0, 0, 0,  4a -  8]
    [0, 1, 0, 0, -4a + 12]          {23*a^2-61*a+68 = 0
    [0, 0, 1, 0,  2a -  6]
    [0, 0, 0, 1,        2]

result[3][1], Display(result[3][2], R);
    [0, 0, 0, 8, 0]
    [1, 0, 0, 0, 0]
    [0, 1, 0, 2, 0]                 {a-4 = 0
    [0, 0, 1, 0, 0]
    [0, 0, 0, 0, 2]

result[4][1], Display(result[4][2], R);
    [0, 0,  4, 0,  0]
    [1, 0, -2, 0,  0]
    [0, 1,  2, 0,  0]                 {a-1 = 0
    [0, 0,  0, 0, -2]
    [0, 0,  0, 1,  0]