I have the equation $$a_i^k g_{kj}=-b_{ij},$$ where $a_i^k$ are given by: $a_i^k \cdot\vec r_k=\vec n_i$.
Here, $g_{ij}$ and $b_{ij}$ are the matrices that determine the first and second fundamental form respectively. So $g_{ij}=\langle \vec r_i,\vec r_j\rangle $ and $b_{ij}=\langle \vec r_{ij},\vec n\rangle$, where $\vec r_{i}=\frac{\partial\vec r}{\partial x_i}$.
My text says: We now multiply both sides with the inverse matrix $g^{jl}$ to get $$a^l_i=-b_{ij}g^{jl}=-b_i^l.$$
Here's what I understand/assume to be true:
- Summation signs are ommited
- $g^{jl}:=(g_{lj})^{-1}$, where $g_{lj}$ is a matrix.
- $i,j,k,l$ all run from $1$ to $n$, so we should have $g_{kj}g^{jl}=\mathrm{Id}_n$
So I say $$a_i^kg_{kj}g^{jl}=\begin{pmatrix}a_i^1g_{11}&\cdots &a_i^1g_{1n}\\ \vdots &\ddots &\vdots\\ a_i^ng_{n1}& \cdots & a_i^ng_{nn}\end{pmatrix} \begin{pmatrix}g_{11}&\cdots &g_{1n}\\ \vdots &\ddots &\vdots\\ g_{n1}& \cdots &g_{nn}\end{pmatrix}^{-1}=\underbrace{\begin{pmatrix}a_i^1& & & \\ & a_i^2& & \\ & & \ddots & \\ & & & a_i^n\end{pmatrix}}_*.$$
So did I calculate $a_i^kg_{kj}g^{jl}$ correctly? I'm not sure, because I assumed the $a_i^k$'s are scalars, so saying $a_i^l=*$, wouldn't make much sense.
I hope it's clear that I am having trouble with the notation that is used.