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Frobenius - inert Frobenius function
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Calling Sequence
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Frobenius(A)
Frobenius(A, 'P')
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Parameters
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A
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square Matrix
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'P'
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(optional) assigned the transformation matrix
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Description
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The Frobenius function is a placeholder for representing the Frobenius form (or Rational Canonical form) of a square matrix. It is used in conjunction with either mod or evala.
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If called in the form Frobenius(A, 'P'), then P will be assigned the transformation matrix corresponding to the Frobenius form, that is, the matrix P such that inverse(P) * A * P = F.
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The call Frobenius(A) mod p computes the Frobenius form of A modulo p which is a prime integer. The entries of A must have rational coefficients or coefficients from an algebraic extension of the integers modulo p.
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The call evala(Frobenius(A)) computes the Frobenius form of the square matrix A where the entries of A are algebraic numbers (or functions) defined by RootOfs.
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Examples
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![A := Matrix([[1+x, 1+x^2], [1+x^2, 1+x^4]])](/support/helpjp/helpview.aspx?si=5044/file00049/math111.png)
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![A := Matrix([[1+x, 1+x^2], [1+x^2, 1+x^4]])](/support/helpjp/helpview.aspx?si=5044/file00049/math114.png)
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![F := Matrix([[0, x+x^5], [1, x+x^4]])](/support/helpjp/helpview.aspx?si=5044/file00049/math121.png)
| (2) |
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![Matrix([[1, 1+x], [0, 1+x^2]])](/support/helpjp/helpview.aspx?si=5044/file00049/math128.png)
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Test the result
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![Matrix([[0, 0], [0, 0]])](/support/helpjp/helpview.aspx?si=5044/file00049/math137.png)
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![A1 := Matrix([[(-3-4*RootOf(_Z^2+1))*x^2+(1-2*RootOf(_Z^2+1))*x-5-4*RootOf(_Z^2+1), (-4+4*RootOf(_Z^2+1))*x^2+(6+3*RootOf(_Z^2+1))*x-6+2*RootOf(_Z^2+1)], [(2+6*RootOf(_Z^2+1))*x^2+(5-3*RootOf(_Z^2+1))*x+2+2*RootOf(_Z^2+1), (-3-5*RootOf(_Z^2+1))*x^2+(4+4*RootOf(_Z^2+1))*x+6+2*RootOf(_Z^2+1)]])](/support/helpjp/helpview.aspx?si=5044/file00049/math141.png)
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![F1 := Matrix([[0, -(1/1145)*(43*RootOf(_Z^2+1)+21)*(-1726*x+442*x^2*RootOf(_Z^2+1)-1482*x*RootOf(_Z^2+1)-1168*x^3*RootOf(_Z^2+1)-2119*x^3+796*x^2-622-144*RootOf(_Z^2+1)+1145*x^4)], [1, -(1/13)*(2+3*RootOf(_Z^2+1))*(39*x^2-16*x+11*x*RootOf(_Z^2+1)+4+7*RootOf(_Z^2+1))]])](/support/helpjp/helpview.aspx?si=5044/file00049/math148.png)
| (5) |
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![Matrix([[1, -(1/25)*(3+4*RootOf(_Z^2+1))*(25*x^2+5*x+10*x*RootOf(_Z^2+1)+31-8*RootOf(_Z^2+1))], [0, (1/5)*(1+3*RootOf(_Z^2+1))*(10*x^2-2*x-9*x*RootOf(_Z^2+1)+4-2*RootOf(_Z^2+1))]])](/support/helpjp/helpview.aspx?si=5044/file00049/math155.png)
| (6) |
Test the result
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![Matrix([[0, 0], [0, 0]])](/support/helpjp/helpview.aspx?si=5044/file00049/math164.png)
| (7) |
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References
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Martin, K., and Olazabal, J.M. "An Algorithm to Compute the Change Basis for the Rational Form of K-endomorphisms." Extracta Mathematicae, (August 1991): 142-144.
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Ozello, Patrick. "Calcul Exact des Formes de Jordan et de Frobenius d'une Matrice." PhD Thesis, Joseph Fourier University, Grenoble, France, 1987.
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