What is the space complexity of calculating Eigenvalues?
The decision versions of many common problems in linear algebra over the integers (or rationals)
are in the class $\mathsf{DET}$, see the paper 

Gerhard Buntrock, Carsten Damm, Ulrich Hertrampf, Christoph Meinel: Structure and Importance of Logspace-MOD Class. Mathematical Systems Theory 25(3): 223-237 (1992)

$\mathsf{DET}$ is contained in $\mathsf{DSPACE}(\log^2)$.

Computing the eigenvalues is a little more delicate:

1) In $\mathsf{DSPACE}(\log^2)$, one can compute the coefficients
of the characteristic polynomial.

2) Then you can use the parallel algorithm by Reif and Neff
to compute approximations to the eigenvalues. The algorithm runs on
a CREW-PRAM in logarithmic time with polynomially many processors, 
so it can be simulated with poly-logarithmic space. 
(It is not explicitely stated in the paper, but their PRAM should to be
log-space uniform.) 
The space used is
 polylogarithmic in the size of
the input matrix and the precision $p$. Precision $p$ means that
you get approximations up to an additive error of $2^{-p}$.

This is a concatenation of functions computable in poly-logarithmic space.
(Output tapes are write only and oneway.)

C. Andrew Neff, John H. Reif: An Efficient Algorithm for the Complex Roots Problem. J. Complexity 12(2): 81-115 (1996)