Let $M$ be a matrix which has the following properties:1) $M$ is Hermitian2) $M$ has only rational entries3) $M$ is known to have rational eigenvaluesWhat algorithms are there for exactly computing the eigenvalues of $M$?Mathematica seems to be able to diagonalise such matrices very very quickly, so such methods must exist (I am hoping they are not trade secrets of Wolfram Inc!). I am sceptical that Mathematica is using the rational roots test on the characteristic equation since the numbers you'd need to factorise get large quickly.
What algorithms are known for computing exact eigenvalues for rational matrices?
linear algebra;algorithms;eigenvalues