We have the squared matrix $A$.
We know this
$$Q^T A Q = D$$ $$S^{-1} A S = D$$
where $D$ is a diagonal matrix, $Q$ orthogonal matrix and $S$ is related with $Q$ by this:
$$S=\alpha Q$$
where $\alpha \in \mathbb{R}$
what I want to know is if we can write $A^{-1}$ in terms of $\alpha$, $S$ and $Q$?
You can't. The diagonal of the matrix $D$ contains the eigenvalues of $A$, while $Q$ represents an orthonormal basis (of the space on which $A$ acts) which diagonalises the action of $A$. There is no way one could recover any information about the actual eigenvalues of a matrix from such a basis alone. Given that $S = \alpha Q$ with $\alpha \in \mathbb{R}$, we have $S^{-1} = \frac{1}{\alpha}Q^{-1} = \frac{1}{\alpha}Q^T$, so the first two equations, $Q^TAQ = D$ and $S^{-1}AS = D$ are equivalent. One can recover $A^{-1}$ in the form $A^{-1} = QD^{-1}Q^T$ from $Q$ and $D$, or equivalently, in the form $A^{-1} = SD^{-1}S^{-1}$ from $S$ and $D$.
