Transpose of a matrix

The transpose of a matrix is a matrix created by reflecting a matrix over its main diagonal, or making the columns rows of the transpose (or vice versa). For example:

$\begin{bmatrix} 3 & 5 & 1 \\ 5 & 6 & 3 \end{bmatrix} ^\mathrm{T} = \begin{bmatrix} 3 & 5 \\ 5 & 6 \\ 1 & 3 \end{bmatrix}$

This can be extended to complex matrices as the conjugate transpose, denoted as ^H. For example:

$\begin{bmatrix} i & 5 - 3i \\ 3 & -1 + i \end{bmatrix} ^\mathrm{H} = \begin{bmatrix} -i & 5 + 3i \\ 3 & -1 - i \end{bmatrix} ^\mathrm{T} = \begin{bmatrix} -i & 3 \\ 5 + 3i & -1 - i \end{bmatrix}$

Notice that for a real matrix, the main diagonal (the diagonal starting the element in the first row and column) remains unchanged. Because of this, the transpose of a real diagonal matrix will simply be the original matrix. More generally, any matrix equal to its own transpose is called a symmetric matrix (or a Hermitian matrix if it is equal to its conjugate transpose).

A matrix equal to the negation of its transpose (i.e. $A=-A^\mathrm{T}$) is called a skew-symmetric matrix.

The most common application of finding the transpose of a matrix is in the process of a finding an inverse of a matrix.