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 = BT AT
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: