When asked to compute the eigenvalues of the following matrix
m = {{-2, 1, 0, 1, 1, 0, 0, 0, 0, 0},
{1, -2, 1, 0, 0, 0, 0, 0, 0, 0},
{0, 1, -2, 0, 0, 0, 0, 0, 0, 0},
{1, 0, 0, -2, 0, 0, 0, 0, 0, 0},
{1, 0, 0, 0, -2, 1, 0, 0, 0, 0},
{0, 0, 0, 0, 1, -2, 1, 0, 0, 0},
{0, 0, 0, 0, 0, 1, -2, 1, 0, 0},
{0, 0, 0, 0, 0, 0, 1, -2, 1, 0},
{0, 0, 0, 0, 0, 0, 0, 1, -2, 1},
{0, 0, 0, 0, 0, 0, 0, 0, 1, -3}}
Mathematica returns
{-4.02739, -3.86914, -3.40114, -2.83334, -2.49755,
-1.81699, -1.32353, -0.901255, -0.333358, 0.00369045}
Are these decimal approximations provably the actual decimal approximations of the eigenvalues(i.e. up to the same term)? In particular, can I be assured that the only positive eigenvalue is actually $>0$? The matrix is non-singular (Mathematica can check it has $|\operatorname{det}|=1$ and I can prove this using non-computational techniques), so there is no possibility the last eigenvalue is zero. Still, I need to rule out the chance that the last eigenvalue is negative.
Total[Eigenvalues[m] // N]
properly gives-21.
in this case, which means that the numerics worked out reliably enough to each displayed decimal. $\endgroup$