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:
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. ) 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.