I was reading Sean Carroll's Introduction to General Relativity
On Page 12 there is an equation given for defining the Lorentz group as a collection of $4\times 4$ matrices that satisfy
$$ \Lambda^T\eta_{\mu \nu} \Lambda = \eta_{\mu \nu} $$
Now he also presents the same defining condition in terms of tensors and index notation on page 13 as:
$$ \eta_{\rho \sigma} = \Lambda^{\mu'}_{\ \ \ \rho} \eta_{\mu' \nu'} \Lambda^{\nu'}_{\ \ \ \sigma} $$
Problem Statement:
I'm trying to figure out what this statement concretely means and prove that it is equivalent to the matrix definition. Unpacking the summation convention I have:
$$ \Lambda^{\mu'}_{\ \ \ \rho} \eta_{\mu' \nu'} \Lambda^{\nu'}_{\ \ \ \sigma} = \sum_{\mu' = 0}^{4} \left[ \Lambda^{\mu'}_{\ \ \ \rho} \eta_{\mu' \nu'} \Lambda^{\nu'}_{\ \ \ \sigma} \right] = \sum_{\nu'=0}^{4} \left[ \sum_{\mu' = 0}^{4} \left[ \Lambda^{\mu'}_{\ \ \ \rho} \eta_{\mu' \nu'} \Lambda^{\nu'}_{\ \ \ \sigma} \right] \right] = \\$$
$$ \left(\Lambda^{0}_{\ \ \ \rho}\eta_{0,0}\Lambda^{0}_{\ \ \ \sigma} + \Lambda^{1}_{\ \ \ \rho}\eta_{1,0}\Lambda^{0}_{\ \ \ \sigma} + ... \right) + \left(\Lambda^{0}_{\ \ \ \rho}\eta_{0,1}\Lambda^{1}_{\ \ \ \sigma} + \Lambda^{1}_{\ \ \ \rho}\eta_{1,1}\Lambda^{1}_{\ \ \ \sigma} + ... \right) + ... $$
Now I don't know how to evaluate $\Lambda^{\beta}_{\ \ \ \alpha} $ Does this refer to the element in row $\beta$ column $\alpha$ or in column $\alpha$ row $\beta$ assuming $\Lambda$ is a $n\times n$ matrix. And generally speaking if I encounter something like $$\Lambda^{i_{0,0} i_{0,1} ... }_{\ \ \ \ \ \ \ \ \ \ \ \ \ i_{1,0} i_{1,1} ...} $$
How do I figure out which element of $\Lambda$ I should be accessing/referring to?
Work attempted:
This person: Definition of the Lorentz group was reading the exact same page, but the OP and answeres did not clarify exactly what was meant by the tensor notation, if someone needs to convert back to matrix notation.
Generally if $A$ is a matrix, I usually take $A_{\alpha, \beta}$ to refer to row $\alpha$, column $\beta$.
