# What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues.

About a year ago, I came across the following way of representing tensors of rank $n$ in matrix form, in a way similar to block matrices. Maybe this is just an alternative to block matrices actually.

Consider a simple $2 \times 2$ matrix definition:

I can represent a $2 \times 2 \times 2$ rank-3 tensor as a cuboid, but this is rather hard to show in a paper or proof, so a convenient notation exists that "splits up" the layers of the cuboid into $2 \times 2$ matrices, which can be flattened out and shown contiguously as follows:

Ditto a $2 \times 2 \times 2 \times 2$ rank-4 tensor can be represented as follows:

For both of these equalities, the notation on the left of the $=$ sign is the standard way of writing block matrices. However the explicit partition on the right is what I'm after: is there a particular name for this notation? Is it actually used somewhere to represent cuboids and other higher rank tensors or did I imagine it?

Thanks in advance if anyone has any info!

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My first guess would be calling your "matrix" on the left a block-matrix or a matrix in block form. That's usually the term I use when they come up when trying to work stuff out in representation theory for instance. But I use this term informally ; I don't know if there's actually a standard term for this. – Patrick Da Silva Dec 14 '12 at 22:30