# What can we say about $n\times n$ invertible matrices, all elements of which are $+1$, $-1$, or $0$?

Basically, the title says it all, except for why I am asking. I'm studying a paper that I can't do justice to in a few words here. (It is not freely available on the Web, as far as I know, but I can email a transcription to anyone interested. For anyone with access to a library that carries IEEE Transactions on Power Apparatus and Systems, the reference is PAS-87:815-824(1968). This was the last paper the author published before he died in 1968.) Suffice it here to say that the author makes up such square matrices out of two rectangular matrices by pasting them together along their conformable edges. He calls this operation a "direct sum", but this is not like any direct sum I ever saw defined in mathematics. The only place I have ever seen such an operation is in the programming language, APL, where it is called "lamination".

After inverting the square matrix, he splits it up again into rectangular matrices of the same shapes as those he started with... and he gives a geometric interpretaion of all four rectangular matrices in terms of the topology of fiber bundles, which I don't understand. Nevertheless, I find this all very fascinating, especially since he applies it to electrical circuits and electromagnetic theory. I suspect that such matrices may be $3$-matroids, but that is another subject that I don't understand.

So, basically, I am on a fishing mission for whatever clues I can come up with here! Does anyone know of a recognized matrix operation like "lamination"? If so, what is it called and in what sort of applications is it used?

I'd really like to communicate more directly with someone knowledgeable in these areas.

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It will be very helpful if you mentioned which paper. Some people may be able to access it even though it's not freely available, and other people may go to the library. Yet other people would be able to tell a lot from the name of the paper and from its author. – Yuval Filmus Sep 2 '11 at 22:01
@Yuval: The title is "Four Abstract Reference Frames of an Electric Network". Through a supporting institution, this link worked for me. – robjohn Sep 2 '11 at 22:42
Sorry, but I see that someone already found it. – Koilon Sep 3 '11 at 2:29