I dont' understand why Sage is unable to give an exact expression for the
eigenvalues of the following matrix :
sage: A= matrix([[0,1],[1,-2]])
sage: [a for a,_,_ in A.eigenvectors_right()]
[-2.414213562373095?, 0.4142135623730951?]
The characteristic polynomial is very simple :
sage: A.charpoly()
x^2 + 2*x - 1
and Sage is able to compute the roots in exact form :
sage: (x^2 + 2*x - 1).roots()
[(-sqrt(2) - 1, 1), (sqrt(2) - 1, 1)]
sage:
What is very surprising is that Sage does recognise the exact values :
sage: L=(x^2 + 2*x - 1).roots(multiplicities=False);L
[-sqrt(2) - 1, sqrt(2) - 1]
sage: [a for a,_,_ in A.eigenvectors_right()]==L
True
sage:
Is there any way to convert the charpoly result to a symbolic expression ?
Note that Maple gives the result in symbolic form:
> restart;with(LinearAlgebra):
>
> A:=Matrix([<0,1>,<1,-2>]);
[0 1]
A := [ ]
[1 -2]
> Eigenvalues(A);
[ 1/2]
[-1 + 2 ]
[ ]
[ 1/2]
[-1 - 2 ]
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