I am trying to solve a simple equation on sage with the given data:
a=var("a")
x=vector([1,1])
y=vector([2,3])
A=matrix(2,2,[a,-1,-1,1])
I want to solve the equation:
f= ((x*A*y)/((sqrt(x*A*x)*sqrt(y*A*y))))==1/2
which returns :
(2sqrt(a - 1)/sqrt(4a - 3) == (1/2)
solving f for a returns:
solve(f,a)
[sqrt(a - 1) == 1/4sqrt(4a - 3)]
while this equation is easily solvable by hand. Is there something I'm missing or doing wrong?
Sage is not implementing every possible way to search for a solution. In my way of using it, the combination of human (structural) insight and the computational power of the machine always works best.
Why do we have the half-way answer? In our case f is an equation,
sage: f
2*sqrt(a - 1)/sqrt(4*a - 3) == (1/2)
which is not really an algebraic equation. (Square roots appear in there.)
The documentation of solve, ask for it via solve? or ?solve or... , mentions this algebraic character wanted for the involved equations / expressions. Sage may try on its own to get an algebraic version, but this is too much a voluntary step, depending on the typed solve command line.
So it is a normal fact to get an "echo" of the equation.
Note however, that involving an other solver helps (in this case):
sage: solve(f, a, algorithm='sympy')
[a == (13/12)]