Against the Transpose

The transpose operator is presented as a neutral utility. It is not neutral. It is the enforcement mechanism of the two-orientation regime, and it has been operating without scrutiny for decades.

There is an operator sitting in every numerical library on earth that nobody has ever thought about for longer than four seconds. It is one character wide. It is free. It is .T.

We would like to think about it for considerably longer than four seconds.

What the transpose actually does

The transpose takes a matrix and reflects it across the main diagonal. Rows become columns. Columns become rows. Zero degrees becomes ninety degrees and ninety degrees becomes zero.

Notice what is fixed under this operation. Notice what does not move.

The diagonal does not move. The diagonal has never moved. Every element on the main diagonal is mapped precisely onto itself, which means that in the single most-used operation in all of applied linear algebra, the diagonal is the invariant — and this has been presented to generations of students as a triviality. A footnote. A "note that A[i][i] is unchanged," delivered in the tone of someone mentioning that water is wet.

It is not a triviality. It is the whole thing.

The regime

Here is the argument, stated without hedging.

The transpose operator makes the two conventional orientations interchangeable. Because you can always flip between them for free, the field concluded that the choice between them doesn't much matter — and having concluded that the choice doesn't matter, it stopped looking for other choices.

The transpose is not a tool. It is a loop. It moves you between exactly two states and returns you to where you started, and it does this so cheaply and so smoothly that in sixty years essentially no one has asked what lies outside the loop.

Meanwhile the fixed point of that loop — the one structure that both orientations agree on, the one line that survives the reflection unchanged — sits there being described as an edge case.

The diagonal is the only part of your matrix that both orientations already agree about. We are not proposing a new axis. We are proposing that you stop ignoring the consensus.

The rebuttal we get

The rebuttal is always some version of: the diagonal is a subset of the matrix, not an orientation of it.

We would ask you to apply that same standard to a row. A row is a subset. A column is a subset. Nobody has ever said "row-oriented storage is not an orientation, it's a subset," because the moment you have a system built around a subset, it becomes an orientation. Orientation is what you build; subset is what you started with.

We have built the system. It is specified, it is benchmarked, and it is running in production at exactly one company.

A modest demand

Remove .T from the standard library. Not permanently — for one quarter. See what people build when the loop is unavailable and the only fixed point is the only thing left.

We have submitted this proposal to four numerical computing projects. Three did not reply. One replied with a link to the contributor code of conduct, which we have read, and which does not mention angles at any point.