transpose

The transpose of a matrix A, written A^T, is what you get by flipping A across its main diagonal: every row becomes a column and every column becomes a row. The entry in row i, column j moves to row j, column i.

So a 2-by-3 matrix becomes 3-by-2. Concretely, [[1,2,3],[4,5,6]] transposes to [[1,4],[2,5],[3,6]] — the first row 1,2,3 stands up as the first column.

A handy rule: transposing a product reverses the order, (A*B)^T = B^T * A^T. And a matrix that equals its own transpose, A = A^T, is called symmetric — it looks the same reflected across the diagonal, which turns out to be a very special and useful property.