Transpose of a matrix

For an $m\times n$ matrix $A$ , its transpose $A^{\mathsf T}$ is the $n\times m$ matrix defined by

(A^{\mathsf T})_{ij}=A_{ji}.

Transposition interchanges rows and columns and satisfies $(A^{\mathsf T})^{\mathsf T}=A$ .

Algebraic identities

Over a commutative field, $(AB)^{\mathsf T}=B^{\mathsf T}A^{\mathsf T}$ , $(A+B)^{\mathsf T}=A^{\mathsf T}+B^{\mathsf T}$ , and $(cA)^{\mathsf T}=cA^{\mathsf T}$ , whenever the operations have compatible sizes.

For complex matrices, the transpose does not conjugate entries. The conjugate transpose is $A^*=\overline A^{\mathsf T}$ ; these two operations agree for real matrices.