I have to prove an equality between matrices $R=OTDO$ where
- $R$ is a $M\times M$ matrix
- $O$ is a $2\times M$ matrix
- $T$ is a $M\times M\times M$ tensor
- $D$ is a diagonal $2\times 2$ matrix
The entries of the matrices and the tensor are probabilities so the result should somehow be the consequence of Bayes formula. The problem is that I have no idea how to compute that because I don't know how to use tensors. I had an algebra course about tensor products of vector spaces a long time ago but it was very abstract so I don't know how to multiply tensors in practice.
I'm surprised because the first matrix of the product has $2$ rows, the last one has $M$ columns and yet the result is a $M\times M$ matrix.
Could you explain how to do this? For example, what's the dimension of $OT$? I am familiar with the Kronecker product of matrices, is it useful here?
Stupid me... I've spent hours trying to understand this product and... this was a typo. It was $O^TDO$ and the equality was straightforward... I've been confused by the fact that the tensor $T$ did exist and there could be and equality involving it. At least I've learnt a few things about tensors, thank you again for the answers!