I made a quick little picture to explain my problem. I would like to figure out how the matrices H and M are related to each-other in the picture below, so that if I was given M, I could calculate H or vice versa:
I have a basis $\left\{e_1, e_2 \right\}$ and its dual basis $\left\{ e^1, e^2 \right\} $, defined in the usual way so that $e^i e _j = \delta^i_j$.
I can write:
$$ V = v_x i + v_y j = v^1 e_1 + v^2 e_2 $$
So then:
$$ \begin{bmatrix} v_x \\ v_y \end{bmatrix} = \underbrace{\begin{bmatrix} \vert & \vert \\ e_1 & e_2 \\ \vert & \vert \end{bmatrix}}_{H^{-1}} \begin{bmatrix} v^1 \\ v^2 \end{bmatrix} $$
So we can do the same thing for the other basis and get:
$$ \begin{bmatrix} v_x \\ v_y \end{bmatrix} = \underbrace{\begin{bmatrix} \vert & \vert \\ e^1 & e^2 \\ \vert & \vert \end{bmatrix}}_{M^{-1}} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} $$
The Relationship
Now my confusion is whether or not this is a valid way to calculate M given H. If I write these to matrices like this:
$$ \underbrace{\begin{bmatrix} – & e_1 & – \\ – & e_2 & –\end{bmatrix}}_{(H^{-1})^T} \underbrace{\begin{bmatrix} \vert & \vert \\ e^1 & e^2 \\ \vert & \vert \end{bmatrix}}_{M^{-1}} = \underbrace{\begin{bmatrix} e_1 \cdot e^1 & e_1 \cdot e^2 \\ e_2 \cdot e^1 & e_2 \cdot e^2 \end{bmatrix}}_{I} $$
The last step is just the identity by definition because of the definition of the basis. So then some simple calculations give us:
$$ I = (H^{-1})^T M^{-1} = (H^{T})^{-1} M^{-1} $$
In the last step, I used the fact that the transpose of the inverse is the inverse of the transpose. So then finally:
$$ H^T = M ^{-1} $$
My Questions
- First off, is this even correct; because I couldn't find any reference for this online.
- If it is correct, does this relationship have a name, and is there some more information on it?
- Is there an intuitive reason why these matrices are related in this way?