Thanks Chris. At most I can add aij to be real positive, the computation
may still take unreasonable of time. I guess this task is unrealistic and
numerical solution could be the only way.
On Wednesday, January 21, 2015 at 5:09:19 PM UTC-5, Chris Smith wrote:
>
> Not only make your variables real, give them a sign (positive, for
> example: var('x', positive=True) is real and positive). This might "still"
> not be enough to allow the sign to be determined, however...I kind of doubt
> that it will, but that is a step in the right direction.
>
> On Wednesday, January 21, 2015 at 3:29:58 PM UTC-6, Junwei Huang 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
>> -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.
>>
>
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