Components of contravariant 2-tensor in new bases

Question

On page 60 of that pdf it says ${\widetilde S}^{ij}=\Lambda_k^i \Lambda_l^j S^{kl}$ as equation (5.3). And then right afterwards there is an exercise that says in terms of matrices (5.3) translates into $$\widetilde S = \Lambda^tS\Lambda$$ The transpose superscript comes before the symbol in the pdf.

Is this correct? Because it seems to me that it should be $$\widetilde S = \Lambda S\Lambda^t$$ because moving the last lambda in the first equation gives ${\widetilde S}^{ij}=\Lambda_k^i S^{kl} \Lambda_l^j$ so that the last lambda should be transposed so that the $l$ variables match correctly.

And moreover it would seem to me that this also matches the expectation in comparison with a linear transformation. In a linear transformation the matrix after change of basis is $\Lambda S L$ where $\Lambda = L^{-1}$. The application of the inverse appears on the left side without any transpose.

Is it in general true that the contravariant (upper) indices transform by something resembling matrix multiplication on the left by $L^{-1}$ and covariant indices transform by something resembling multiplication on the right. Because to me ${\widetilde S}^{ij}=\Lambda_k^i \Lambda_l^j S^{kl}$ looks awfully like multiplying on the left twice but of course its not equal to $\Lambda \Lambda S$ so we can't say its matrix multiplication on the left twice.

Answer 1

${\widetilde S}^{ij}=\Lambda_k^i \Lambda_l^j S^{kl}$ as equation (5.3). And then right afterwards there is an exercise that says in terms of matrices (5.3) translates into $$\widetilde S = \Lambda^tS\Lambda$$ The transpose superscript comes before the symbol in the pdf.

Ok, so on page 19 of the pdf the author publishes his conventions for matrix indices and multiplication: $$ (AB)^i_j = A^i_kB^k_j$$ this means $i$ is the row-index whereas $j$ is the column index and of course $k$ is summed over ala Einstein. Considering your expression, $$ \Lambda_k^i \Lambda_l^j S^{kl} $$ there is a free upper $ij$ set of indices. This means it is not the components of a matrix of a linear transformation. Instead, this is the matrix of some mapping which has an expression like: $$ \widetilde S(\alpha, \beta) = \widetilde S^{ij}\alpha_i\beta_j $$ In my thinking, this would be a bilinear form on the dual space. That said, if we wish, we can define matrix operations on such matrices by ignoring the up/down conventions for the sake of convenience: again consider: $$ \Lambda_k^i \Lambda_l^j S^{kl} = \Lambda_k^i S^{kl} \Lambda_l^j $$ if $\Lambda_k^i$ is to have row-index $i$ and column-index $k$ then it follows that $\Lambda_l^j$ also has row-index $j$ and column-index $l$. But, this is no good for matrix multiplication since we have to contract a column-index of the left factor against a row-index of the right factor (e.g. $(AB)^i_j = A^i_kB^k_j$). Hence, we need to swap out $\Lambda$ for $\Lambda^T$ where $(\Lambda^T)^l_j = \Lambda^j_l$ then $$ \Lambda_k^i \Lambda_l^j S^{kl} = \Lambda_k^i S^{kl} (\Lambda^T)^l_j = (\Lambda S \Lambda^T)^{ij} $$ So it appears I agree with you. But, in fairness to the author of the pdf, I have not read all of it and it is entirely possible I fail to appreciate another convention at play here. My larger comment would be this: matrices are a poor notation in such contexts. The up-down index notation more manifestly reveals the true coordinate-change nature of the object in question.

In particular, if $V$ is a vector space of finite dimension over $\mathbb{R}$,

1. $V \times V \rightarrow \mathbb{R}$ maps can be written as $T(x,y) = T_{ij}x^iy^j$ so the "matrix" of such a map has components $T_{ij}$,
2. $V^* \times V^* \rightarrow \mathbb{R}$ maps can be written as $T(\alpha,\beta) = T^{ij}\alpha_i\beta_j$ so the "matrix" of such a map has entries is $T^{ij}$,
3. $V \times V^* \rightarrow \mathbb{R}$ maps can be written as $T(x,\alpha) = T_{i}^{\ j}x^i\alpha_j$ thus the "matrix" of such a map has components $T_{i}^{\ j}$,
4. $V^* \times V \rightarrow \mathbb{R}$ maps can be written as $T(\alpha,x) = T^{i}_{\ \ j}\alpha_ix^j$ so the "matrix" of such a map has components $T^{i}_{\ \ j}$.

So, how does a matrix change under coordinate change? I say forget that question and focus instead on tensors presented by index formulas. Matrix multiplication merely muddies such discussions unless one is very careful to announce conventions. Even so there is likely some abuse of conventions needed.