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ComprehensiveRank.md

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ComprehensiveRank

The ComprehensiveRank function will compute a complete case discussion of the rank of a matrix of multivariate polynomials.

Calling Sequence

ComprehensiveRank(A, R, options)
ComprehensiveRank(A, rs, R, options)
ComprehensiveRank(A, cs, R, options)
ComprehensiveRank(A, F, R, options)
ComprehensiveRank(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.

Output

Output structure Options
[r, rs] 'outputType' is 'RegularSystem' or 'RS'
[r, cs] 'outputType' is 'ConstructibleSet' or 'CS'

Where r is the rank of the input matrix for all parameter values in the zero set of cs or rs. 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([[1, 2, 3],
             [4, 5, 6],
             [7, 8, a]]):

# Compute the rank of A
result := ComprehensiveRank(A, R):

# Print the result
result[1][1], Display(result[1][2], R);
    2, a - 9 = 0

result[2][1], Display(result[2][2], R);
    3, a - 9 <> 0