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RealComprehensiveRank.md

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RealComprehensiveRank

The RealComprehensiveRank function will compute a complete case discussion of the rank of a matrix where the entries are multivariate polynomials whose indeterminants are treated as parameters. Parameters are assumed to be real valued and computation is done over polynomial equality, inequation and inequality (strict and non-strict) constraints on the parameters.

Calling Sequence

RealComprehensiveRank(A, R)
RealComprehensiveRank(A, rsas, R)
RealComprehensiveRank(A, lrsas, R)
RealComprehensiveRank(A, F, R)
RealComprehensiveRank(A, F, H, R)
RealComprehensiveRank(A, F, N, P, H, R)

Input

Variable Description
A Square matrix of multivariate polynomials over R
rsas Regular semi-algebraic system
lrsas List of regular semi-algebraic systems
F List of polynomials over R representing equality constraints on the parameters
F List of polynomials over R representing equality constraints on the parameters
N List of polynomials over R representing non-negativity constraints on the parameters
P List of polynomials over R representing positivity constraints on the parameters
R Polynomial ring

Output

A list with elements of the form [r, lrsas] where r is the reank of the input matrix for all parameter values in the zero set of lrsas. Together, all the lists of regular semi-algebraic systems form a partition of the input constraints on the parameters.

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 := RealComprehensiveRank(A, [], [], [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 > 0 and a - 9 <> 0]