On Wed, Jan 21, 2015 at 3:29 PM, Junwei Huang <[email protected]> wrote:
> Hello all
> Is it too greedy to ask computer to give symbolic expression for this
> matrix? I can't get the result either from mathematicaTM or sage.
> a11,a12,a13,a14,a15,a16,a22,a23,a24,a25,a26,a33,a34,a35,a36,a44,a45,a46,
> a55,a56,a66 = var(
> 'a11,a12,a13,a14,a15,a16,a22,a23,a24,a25,a26,a33,a34,a35,a36,a44,a45,a46,a55,a56,a66'
> ,real=True)
> p,q,r=var('p,q,r',real=True)
> g11=a11*p*p+a66*q*q+a55*r*r+2.0*a16*p*q+2.0*a15*p*r+2.0*a56*q*r;
> g22=a66*p*p+a22*q*q+a44*r*r+2.0*a26*p*q+2.0*a46*p*r+2.0*a24*q*r;
> g33=a55*p*p+a44*q*q+a33*r*r+2.0*a45*p*q+2.0*a35*p*r+2.0*a34*q*r;
> g12=a16*p*p+a26*q*q+a45*r*r+(a12+a66)*p*q+(a14+a56)*p*r+(a46+a25)*q*r;
> g13=a15*p*p+a46*q*q+a35*r*r+(a14+a56)*p*q+(a13+a55)*p*r+(a36+a45)*q*r;
> g23=a56*p*p+a24*q*q+a34*r*r+(a46+a25)*p*q+(a36+a45)*p*r+(a23+a44)*q*r;
>
> m = Matrix([[g11,g12,g13],[g12,g22,g23],[g13,g23,g33]])
> m0 = m.subs([(a14,0),(a15,0),(a16,0),(a24,0),(a25,0),(a26,0),(a34,0),(a35,
> 0),(a36,0),(a45,0),(a46,0),(a56,0),(a22,a11),(a23,a13),(a55,a44)])
>
> eigvals0=m0.eigenvals()
> eigvals=m.eigenvals()
>
> For matrix m0, sage works but sympy fails:
> TypeError: cannot determine truth value of
>
This error typically indicates a bug in SymPy.
> -a11**2*a33*p**2*q**2*r**2 - a11**2*a44*p**4*q**2 - a11**2*a44*p**2*q**4 +
> 2*a11*a13**2*p**2*q**2*r**2 + 4*a11*a13*a44*p**2*q**2*r**2 -
> a11*a33*a44*p**2*r**4 - a11*a33*a44*q**2*r**4 - a11*a33*a66*p**4*r**2 -
> a11*a33*a66*q**4*r**2 - a11*a44**2*p**4*r**2 - a11*a44**2*q**4*r**2 -
> a11*a44*a66*p**6 - a11*a44*a66*p**4*q**2 - a11*a44*a66*p**2*q**4 -
> a11*a44*a66*q**6 + a12**2*a33*p**2*q**2*r**2 + a12**2*a44*p**4*q**2 +
> a12**2*a44*p**2*q**4 - 2*a12*a13**2*p**2*q**2*r**2 -
> 4*a12*a13*a44*p**2*q**2*r**2 + 2*a12*a33*a66*p**2*q**2*r**2 -
> 2*a12*a44**2*p**2*q**2*r**2 + 2*a12*a44*a66*p**4*q**2 +
> 2*a12*a44*a66*p**2*q**4 + a13**2*a44*p**2*r**4 + a13**2*a44*q**2*r**4 +
> a13**2*a66*p**4*r**2 - 2*a13**2*a66*p**2*q**2*r**2 + a13**2*a66*q**4*r**2 +
> 2*a13*a44**2*p**2*r**4 + 2*a13*a44**2*q**2*r**4 + 2*a13*a44*a66*p**4*r**2 -
> 4*a13*a44*a66*p**2*q**2*r**2 + 2*a13*a44*a66*q**4*r**2 - a33*a44**2*r**6 -
> a33*a44*a66*p**2*r**4 - a33*a44*a66*q**2*r**4 - 4*a44**2*a66*p**2*q**2*r**2
> + 2*(-a11*p**2 - a11*q**2 - a33*r**2 - a44*p**2 - a44*q**2 - 2*a44*r**2 -
> a66*p**2 - a66*q**2)**3/27 - (-a11*p**2 - a11*q**2 - a33*r**2 - a44*p**2 -
> a44*q**2 - 2*a44*r**2 - a66*p**2 - a66*q**2)*(a11**2*p**2*q**2 +
> a11*a33*p**2*r**2 + a11*a33*q**2*r**2 + a11*a44*p**4 + 2*a11*a44*p**2*q**2
> + a11*a44*p**2*r**2 + a11*a44*q**4 + a11*a44*q**2*r**2 + a11*a66*p**4 +
> a11*a66*q**4 - a12**2*p**2*q**2 - 2*a12*a66*p**2*q**2 - a13**2*p**2*r**2 -
> a13**2*q**2*r**2 - 2*a13*a44*p**2*r**2 - 2*a13*a44*q**2*r**2 +
> 2*a33*a44*r**4 + a33*a66*p**2*r**2 + a33*a66*q**2*r**2 + a44**2*p**2*r**2 +
> a44**2*q**2*r**2 + a44**2*r**4 + a44*a66*p**4 + 2*a44*a66*p**2*q**2 +
> a44*a66*p**2*r**2 + a44*a66*q**4 + a44*a66*q**2*r**2)/3 < 0
>
> For matrix m, I guess it just takes a bit longer to fail.
>
> I just want the same eigenvalue associated with the same eigenvector or
> eigensystem as aij, p,q,r vary. If I use numerical solution I lose the
> track of the eigensystem. Any alternatives, or suggestions? Thanks very
> much.
>
The general eigenvalues of a 3x3 matrix involve the roots of a cubic
polynomial, which involve a somewhat nasty equation. It will also be
difficult for SymPy to manipulate these eigenvalues into finding the
eigenvectors.
It might help to make g11, g12, ... into Symbols, find the eigenvectors of
the matrix of symbols, and replace them with their full values as late as
possible.
Aaron Meurer
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