Matrices as second order tensors proof?

I am trying to proof that all matrices are tensors.

I have got to a stage where I need to proof that:

$$\gamma_{li} \gamma_{kj}= \frac{\partial q_j}{\partial q_k'} \frac{\partial q'_l}{\partial q_i}$$ Where $\gamma_{ij}=\frac{h_i}{h'_j} \frac{\partial q_i}{\partial q'_j}$ (and $h_i$ is the scale factor for $q_i$ and $h'_i$ is the scale factor for $q'_i$).

I can do this when the primed and unprimed coordinates are both Cartesian, but not if they are not. So can the above equation be proved in general, if so how (a source would be helpful) and if not why not?

Answer

I am trying to proof that all matrices are tensors.

Matrices are not tensor, rather finite dimensional representations thereof, which therefore transform accordingly. Given $V$ as a vector space and $V^*$ as its dual, a tensor of type $(r,s)$ is, by definition, any multilinear map

$$\tau\colon V^r\times{V^*}^s\to \mathbb{F}$$ $\mathbb{F}$ being any field. Chosen a basis $\left\{\textbf{e}_i\right\} \in V$ and its corresponding dual $\left\{\alpha^j\right\} \in V^*$ such that $\alpha^j(\textbf{e}_i)=\delta^j_i$ the components of the tensor with respect to these bases are the values of the multilinear map $\tau$ evaluated thereupon. Standard matrix multiplication rules for components follows by distributivity of the product and the rules for the change of basis.

As an example for $(r,s)=(1,1)$ we have

$$\tau\colon V\times V^*\to\mathbb{R}$$ with $\tau_i^j=\tau(\textbf{e}_i,\alpha^j)$. Given a change of basis as orthogonal matrix $O$ we have $$\tau'(\textbf{e}_i',\alpha'^j)=\tau(O_i^k\textbf{e}_k,\,{O^{-1}}^j_r\alpha^r)= O_i^k\,{O^{-1}}^j_r\,\tau_k^r$$ which terminates the proof.

A similar answer along the same lines can be found here.