I'm having a problem multiplicating 3 matrices in index notation. I know this should be trivial but I just can't figure it out. Is there any formula like
$$\ A'_{\mu\nu} = M_{\mu}^{\ \rho}(M^{-1})_{\nu}^{\ \theta}A_{\rho\theta }$$ $\rightarrow$ $A'=MAM $
(M is diagonal if this changes anything and summation over identical indices is assumed (Einstein summation convention).)
For converting between matrix and index notation?, As far as i know the following relation holds
$\ A'_{\mu \nu} = M_{\mu}^{\ \rho}M_{\nu}^{\ \theta}A_{\rho\theta }$ $\rightarrow$ $A'=MAM^{-1} $
as physicists use it in special relativity, but I can't derive this formula either because I just can't figure out where the inverse matrix comes into play. I of course already wrote down specific examples of the sum on the left hand side and the matrix multiplication on the right hand side, but it always seems to work without the inverse.
Any help, tip or link where the equivalence is shown explicitly will be much much appreciated because I'm stuck with this problem for quite a while.
Best Answer
Matrix multiplication with non-raised (i.e., not written as upper or lower) indices, the first index being the row index and the second the column index, is given by the rule $$ (AB)_{i,k}=\sum_jA_{i,j}B_{j,k}\tag1 $$ Now your second rule for transforming $A$ to $A'$ can be written (if you'll forgive me for using non-Greek letters as indices) $$ A'_{i,l}= \sum_{j,k} M_i^jA_{j,k}M_l^k,\tag2 $$ (I've inverted the indices in the LHS since I think you made a mistake: in your formula, if $M$ is the identity then $MAM^\top$ switches the indices, which cannot be right; with this proviso the correspondence is $\mu:=i$, $~\rho:=j$, $~\theta:=j$, $~\nu:=k$). Now if we agree to call the lower index of $M$ the first one and the upper index the second one, then in the right hand side of $(2)$, the second copy of $M$ has its indices switched with respect to what one would get by expanding out $MAM$ using $(1)$. So to get the indices in the right place one must transpose the second copy of $M$ before entering it into the matrix product: the RHS of $(2)$ describes the computation of $MAM^\top$.