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ComprehensiveSmithForm

The ComprehensiveSmithForm function will compute a complete case discussion of the Smith normal form of a matrix of monic, parametric univariate polynomials.

Calling Sequence

ComprehensiveSmithForm(A, v, R, options)
ComprehensiveSmithForm(A, v, rs, R, options)
ComprehensiveSmithForm(A, v, cs, R, options)
ComprehensiveSmithForm(A, v, F, R, options)
ComprehensiveSmithForm(A, v, F, H, R, options)

Input

Variable Description
A Square matrix of monic, parametric univariate polynomials in v
v The variable that the entries of A are monic polynomials in
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 where v is the greatest variable

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' 'S' Which matrices to return, value can be either 'S', 'U', or 'V' or a list containing any combination of 'S', 'U', and 'V'
'lazy' false When true, this method will only return one branch of the computation. This option is not currently implemented.
'algorithm' 'minors' Specify which algorithm to use, currently the 'minors' algorithm is the only one implemented.

Option Compatibility

  • Only the Smith form can be returned when the minors algorithm is used. The transformation matrices can not be computed with this algorithm.

Output

A list with elements in one of the following forms:

Output structure Options
[S, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = 'S'
[S, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = 'S'
[U, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = 'U'
[U, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = 'U'
[V, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = 'V'
[V, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = 'V'
[S, U, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = ['S', 'U']
[S, U, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = ['S', 'U']
[S, V, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = ['S', 'V']
[S, V, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = ['S', 'V']
[S, U, V, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = ['S', 'U', 'V']
[S, U, V, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = ['S', 'U', 'V']
[U, V, rs] 'outputType' is 'RegularSystem' or 'RS' and 'outputMatrices' = ['U', 'V']
[U, V, cs] 'outputType' is 'ConstructibleSet' or 'CS' and 'outputMatrices' = ['U', 'V']

Where S is the Smith form of the input matrix for all parameter values in the zero set of cs or rs. U and V are the left and right transformation matrices respectively such that S = U.A.V. Together, all the constructible sets or regular systems form a partition of the input constructible set, regular system or set of equations and inequations.

Assumptions

  1. v is the greatest variable of R
  2. A is a square matrix
  3. All entries of A are monic in v
  4. F and H do not contain any polynomials containing v
  5. If a constructible set or regular system is input, v cannot be a variable in any equations or inequations

Example

with(RegularChains):
with(ParametricMatrixTools):

# Set up the matrix
R := PolynomialRing([x,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]]):
Ax := x*IdentityMatrix(5) - A:

# Compute the Smith form of Ax
result := ComprehensiveSmithForm(Ax, x, [], R):

# Print the result
map(factor, result[1][1]), Display(result[1][2], R);
    [1,0,0,0,0]
    [0,1,0,0,0]                              {a-4 <> 0
    [0,0,1,0,0]                              {a-1 <> 0
    [0,0,0,1,0]                              {23a^2-61a+68 <> 0
    [0,0,0,0,-(x-2)(x^2+2)(-x^2+2a-4)]

map(factor, result[2][1]), Display(result[2][2], R);
    [1,0,0,0,0]
    [0,1,0,0,0]
    [0,0,1,0,0]                              {23a^2-61a+68 = 0
    [0,0,0,1,0]
    [0,0,0,0,-(x-2)(x^2+2)(-x^2+2a-4)]

map(factor, result[3][1]), Display(result[3][2], R);
    [1,0,0,0,  0]
    [0,1,0,0,  0]
    [0,0,1,0,  0]                     {a-4 = 0
    [0,0,0,x-2,0]
    [0,0,0,0,  (x-2)*(x+2)*(x^2+2)]

map(factor, result[4][1]), Display(result[4][2], R);
    [1,0,0,0,    0]
    [0,1,0,0,    0]
    [0,0,1,0,    0]                     {a-1 = 0
    [0,0,0,x^2+2,0]
    [0,0,0,0,    (x-2)*(x^2+2)]