# Conjugate transpose

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication:

Thus, an *m*-by-*n* matrix of complex numbers could be well represented by a 2*m*-by-2*n* matrix of real numbers. The conjugate transpose therefore arises very naturally as the result of simply transposing such a matrix—when viewed back again as *n*-by-*m* matrix made up of complex numbers.