From Wikipedia, the Minkowski metric is defined (using (- + + +) signature) as : $$\eta_{\mu \nu} = \eta^{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
I am a beginner in tensor calculus, and feel uneasy that a covariant vector is being equated to a contravariant vector. Is it then possible to simply replace the covariant Minkowski metric tensor by its contravariant counterpart withing a calculation ?
For example, given a four vector $k$, is it correct that :
$k \cdot k = \eta_{\alpha \mu}k^{\mu} k^{\alpha} = \eta^{\alpha \mu}k^{\mu} k^{\alpha}$
Are there any problems with summing over all upper indices ?