Matrix transpose

A matrix transpose is a swap of the row and column indices:

Why is this such a useful operation? The key is to think of it as a kind of reciprocal: instead of thinking of all the ‘s for each , we are thinking about the ‘s for each . This is very much not the same as a matrix inverse, even though it might feel like it should be. (Matrices for which are called orthogonal matrices).

There are many situations where you’ll want to use both a matrix and its transpose in a single operation. This is a bit like first looking something up in a spreadsheet by column, and then using that information to look up something else by row.