- On the Transpose of Linear Operators

1. The Relationship Between $A$ and $A^T$ #

$$ e_i = A e_{i’} = A_i^{j’} e_{j’} $$

Meanwhile,

$$ e_j(e^{i’}) = e^{i’} (e_j) = e^{i’} (A_j^{k’} e_{k’}) $$

$$ = A_j^{k’} e^{i’} (e_{k’}) = A_j^{k’} \delta_{i’k’} = A_j^{i’} $$

Therefore, $e^{i’} = A_j^{i’} e^j$ holds, which can be interpreted as $(e’)^i = \sum_j (A^T)_i^{j} e^j = A^T e^i$.

Consequently,

$$ A : W \to V,\quad e_{i’} \mapsto e_i \quad $$

$$ \leftrightarrow \quad A^T : V^* \to W^*,\quad e^i \mapsto e^{i’} $$

This relationship holds. Here, we are addressing the case where W=V.

2. Additional Remarks #

Through $e_i = A_i^{j’} e_{j’}$ and $e^{i’} = A_j^{i’} e^j$, we can confirm that the Schouten convention works very well. Additionally, both $e_i$ and $e^i$ are represented as column vectors, and in matrices, $A_i^j$ indicates that i represents the column and j represents the row in both vector spaces and dual spaces.