Connection between Matrices with Different Index Configurations in Special Relativity I am studying special relativity and I can't figure out what is the difference between the matrix-index notation between:
$$    Λ_{α}{}^{β}, Λ^{α}{}_{β}, Λ^{αβ} ,Λ_{αβ} $$
Why do we introduce this kind of notation in SR? In Linear Algebra, we denote matrices simply with $  Λ_{ij} $ where $i,j$ are indices that run in some common (or not) set. I know that the answer is somehow related with covariance and contravariance (which is important in SR mathematical formalism) but I don't know how exactly.
 A: For the sake of concreteness and without going into details of covariant and contravariant indices, let's just show the difference with the help of an example:
$\Lambda^\alpha_{\,\,\beta}$ is usually used to denote a Lorentz transformation. Let's take a transformation in $x$-direction:
$$\Lambda=\Lambda^\alpha_{\,\,\beta}=\left(\begin{array}{c c c c}\gamma  & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &\gamma\end{array}\right)$$
When this transformation is applied to a four-vector $x^\beta=\left(\begin{array}{c}x\\y\\z\\t\end{array}\right)$, i.e. $\Lambda\cdot x=\Lambda^\alpha_{\,\,\beta} x^\beta
$, one obtains the standard Lorentz transformation $x^\prime = \gamma(x-vt)$, $y^\prime = y$, $z^\prime = z$, and $t^\prime = \gamma(t-\frac{v}{c^2}x)$.
Now, it is easy to show the difference between the different index notations: $$ \Lambda_{\alpha\beta}= \eta_{\alpha\gamma}\Lambda^\gamma_{\,\,\beta}=\left(\begin{array}{c c c c}1 & 0 & 0 &0 \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\0& 0 & 0 &-c^2\end{array}\right)\cdot \left(\begin{array}{c c c c}\gamma  & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &\gamma\end{array}\right)=\left(\begin{array}{c c c c}\gamma  & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\\gamma v & 0 & 0 &-\gamma c^2\end{array}\right)$$
and similarly
 $$ \Lambda^{\alpha\beta}= \Lambda^\alpha_{\,\,\gamma}\eta^{\gamma\beta}=\left(\begin{array}{c c c c}\gamma  & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &\gamma\end{array}\right)\cdot \left(\begin{array}{c c c c}1 & 0 & 0 &0 \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\0& 0 & 0 &-\frac{1}{c^2}\end{array}\right)=\left(\begin{array}{c c c c}\gamma  & 0 & 0 &\gamma \frac{v}{c^2} \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &-\gamma\frac{1}{c^2}\end{array}\right)$$
