# 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?

• This question will be easier to answer if you specify which definition of tensor you're using. Mathematically, a tensor is often defined as a multilinear map from a product of vector spaces/dual spaces into a field, in which case the fact that matrices are tensors follows immediately from the fact that matrix multiplication is distributive. If your definition explains tensors in terms of some transformation law (as it seems you do), it would be helpful to know exactly what rule you're working with in order to provide a useful answer. – user35736 Jul 1 '15 at 17:57
• @user35736 I am using the definition that a type $(m,n)$ tensor is an object that transforms as follow $$\bar T^{i_1,...,i_m}_{j_1,...,j_m} =T^{u_1,...,u_m}_{v_1,...,v_m} (\frac{\partial q'_{i_1}}{\partial q_{u_1}}...\frac{\partial q'_{i_m}}{\partial q_{u_m}})(\frac{\partial q_{v_1}}{\partial q'_{j_1}} ...\frac{\partial q_{v_n}}{\partial q'_{j_n}})$$ – Quantum spaghettification Jul 1 '15 at 18:05
• Are you familiar with the idea of dual vector spaces/dual bases to a standard vector space basis? Tensors are a difficult thing to talk about since they're defined in vastly different ways at different levels of instruction so it's tricky to phrase an answer to a question like this. Do you have a specific reference text you're using to learn this material? – user35736 Jul 1 '15 at 18:25
• @user35736 I am familiar with dual vector spaces etc. But I would prefer to do it using this definition (and to see if it is possible). I am kind of using 'Methods of theoretical physics' by Morse and Feshback books.google.co.uk/… along with a view other books to do with tensors – Quantum spaghettification Jul 1 '15 at 19:24

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.