On Wed, Feb 4, 2015 at 1:14 AM, Chris Smith <[email protected]> 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())
>
That's a true fact. However, if the matrix is positive definite, then (by
definition) all the eigenvalues are positive, so you should be able to take
the square root on both sides. But sqrt(det(M)) !=
sqrt(eigen1)*sqrt(eigen2)*... in general, due to the fact that sqrt(x*y) !=
sqrt(x)*sqrt(y) in general.
Aaron Meurer
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