General

Transposing a matrix turns the columns into rows and vice-versa

Similarly, transposing turns a column vector into a row vector and vice-versa.

Another way of thinking about it is that the elements are flipped over around the ``main diagonal'', which runs from top left to bottom right:

(The sum of the elements on the main diagonal is called the trace of the matrix.)

Note that .

Transpose in index notation:

Note that in index notation, the main diagonal consists of the elements where i=j. These stay put during transposing.

Transposing matrix products:

(AB)^T = B^T A^T

For complex matrices, the normal generalization of transpose is ``Hermitian conjugate'', where you take the complex conjugate of each complex number, in addition to interchanging rows and columns: , or .

Example: