[index]

Algebra::JordanForm

(Jordan Matrix)

This class expresses the Jordan canonical matrix.

File Name:

• jordan-form.rb

SuperClass:

• Object

Included Modules

none.

Associated Functions:

Algebra::MatrixAlgebra#jordan_form

Returns the Jordan matrix of self.

Algebra::MatrixAlgebra#jordan_form_info

Same as Algebra::JordanForm.decompose(self).

Class Methods:

::new(array)

Returns a instance of JordanForm. array is the array of the pair of [diagonal component, size]. To get the matrix expression, apply to_matrix for the upper triaglular matrix or to_matrix_l for the lower.

Exampe:

j = Algebra::JordanForm.new([[2, 3], [-1, 2]])
j.to_matrix.display #=>
#  2,   1,   0,   0,   0
#  0,   2,   1,   0,   0
#  0,   0,   2,   0,   0
#  0,   0,   0,  -1,   1
#  0,   0,   0,   0,  -1

::construct(elem_divs, facts, field, pfield)

abbreviate.

::decompose(m)

Returns the array:

[jm, tL, sR, field, modulus]

where jm is the Jordan matrix of m, tL and sR are the left and right deformation matrix from m tojm, field is the mimimal decompostion field to Jordan decompose and modulus is the polynomials to the algebraic extention. (tL * sR == the identity matrix)

Example:

m = Algebra.SquareMatrix(Rational, a.size)[
 [-1, 1, 2, -1],
 [-5, 3, 4, -2],
 [3, -1, 0, 1],
 [5, -2, -2, 0]
]
jf, p, q, field, modulus = Algebra::JordanForm.decompose(m)
jf.display; puts #=>
#  2,   0,   0,   0
#  0,   a,   0,   0
#  0,   0,   b,   0
#  0,   0,   0, -b - a

p modulus #=> [a^3 + 3a - 1, b^2 + ab + a^2 + 3]

print "P =\n"; p.display; puts
print "P^-1 =\n"; q.display; puts

m = m.convert_to(field)
p jf == p * m * q #=> true

Methods:

to_matrix(ring)

Returns the upper triangular Jordan matrix over ring.

Example:

j = Algebra::JordanForm.new([[2, 3], [-1, 2]])
j.to_matrix(Integer).display #=>
#  2,   1,   0,   0,   0
#  0,   2,   1,   0,   0
#  0,   0,   2,   0,   0
#  0,   0,   0,  -1,   1
#  0,   0,   0,   0,  -1

to_matrix_r(ring)

Same as to_matrix.

to_matrix_l(ring)

Returns the lower triangular Jordan matrix over ring.