Thanks for all the reply !
The reason that I would prefer the answer to be in something like
sqrt(47)/3
is that the answer would be converted to Latex. To me, it just look
nicer
in Latex to have \sqrt(47)/3 than a number with many digits.
Thanks!
Shing
On Sep 21, 6:01 pm, Jason Grout <[email protected]> wrote:
> Shing Hing Man wrote:
> > In Sage, the eigenvalues and vectors (over a Rational matrix) returns
> > the answer in numerical format.
>
> It's not really numerical format. It's the same as the Root objects in
> Mathematica or Maple. Recent versions of Mathematica even print out the
> numeric approximations of the roots; Sage just prints that out the
> numerical approximation by default.
>
> For example:
>
> sage: sageMatrix = matrix(QQ,[[1,4],[4,2]])
> sage: sageMatrix.eigenvalues()
> [-2.531128874149275?, 5.531128874149275?]
> sage: eig=sageMatrix.eigenvalues()
> sage: eig[0]
> -2.531128874149275?
> sage: a=eig[0]
>
> Now, "a" is *not* a numeric value, though it is approximately equal to
> the number displayed above. It is an exact root of a polynomial:
>
> sage: a.minpoly()
> x^2 - 3*x - 14
> sage: a^2-3*a-14
> 0
>
> You see the advantages to displaying things this way once you get
> matrices that have eigenvalues which are roots of 3rd degree or higher
> polynomials. Instead of the huge long "Root[]" displays that
> Mathematica does, Sage just displays the numeric approximation for the
> exact eigenvalue.
>
> There has been discussion of displaying a square root if the exact
> eigenvalue is actually a square root (or, more generally, in technical
> terms, if an element of QQbar has minpoly() or some easily-calculatable
> approximation of minpoly() showing that we have a root of a quadratic,
> then display the square root). However, if you do that, then you still
> have the problem of not knowing approximately what the value is. Which
> is easier to see and make sense of:
>
> sqrt(47)/3
>
> or
>
> 2.285218200133682?
>
> (we get that from:
>
> sage: QQbar(sqrt(47)/3)
> 2.285218200133682?
>
> )
>
> Note that the two are *exactly* equal---there are no numeric issues
> involved.
>
> Would you still like to see sqrt(47)/3 as an eigenvalue, rather than the
> exactly same thing, but printed out as a numerical approximation?
>
> Thanks,
>
> Jason
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