Chris,
Thank you for looking into this! I really appreciate it.
And you are right that det(M) == Mul(*M.eigenvals())
However, if M is positive definite, then sqrt(det(M)) should also equal to
sqrt(lambda1)*sqrt(lambda2)*sqrt(lambda3), where lambda1 - 3 are the
eigenvalues of the matrix. (Because positive-definiteness would mean
det(M)>0, lambda1-3 > 0)
And this is currently where I got stuck...
Shawn
On Wednesday, February 4, 2015 at 2:14:14 AM UTC-5, Chris Smith wrote:
>
> A brief description is - the sqrt(matrix determinant) is not equal to the
>> product of the eigenvalues of matrix**(1/2).
>>
>> Could you please help a little bit in there...? Would what I am doing
>> even be mathematically correct?
>>
>
> >>> var('a:d')
> (a, b, c, d)
> >>> m=Matrix(2,2,[a,b,c,d])
> >>> m.det()
> a*d - b*c
> >>> m.eigenvals()
> {a/2 + d/2 - sqrt(a**2 - 2*a*d + 4*b*c + d**2)/2: 1, a/2 + d/2 + sqrt(a**2
> - 2*a*d + 4*b*c + d**2)/2: 1}
> >>> Mul(*_).simplify()
> a*d - b*c
>
> >>> m3=Matrix(3,3,var('x:9'))
> >>> d=m3.det()
> >>> e=Mul(*m3.eigenvals()).simplify()
> >>> d==e
> True
>
> Does someone have the identity wrong? above it appears that det(M) =
> Mul(*M.eigenvals())
>
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