Transpose ($A^T$)

The matrix transpose interchanges rows and columns: $(A^T{)}_{ij}=A_{ji}$. On a fixed matrix space, $A\mapsto A^T$ is the transposition map, a linear map that preserves the trace norm but is not completely positive in dimensions greater than one.

For real rectangular matrices, expanding $(A+H{)}^t(A+H)$ leaves the quadratic remainder $H^{t}H$, whose matrix norm is $O(\Vert H{\Vert }^2)$. Thus the displayed Fréchet derivative is a linear map in the increment $H$. At an orthogonal matrix, its kernel consists of increments $H=A\Omega$ with $\Omega$ skew-symmetric. This is the tangent-space condition obtained by differentiating $A^{t}A=I$.

For compatible finite matrices, the transpose reverses the factor order. Entrywise, $[(AB{)}^T{]}_{ij}={\sum }_k{A}_{jk}{B}_{ki}={\sum }_k(B^T{)}_{ik}(A^T{)}_{kj}$, proving the identity. The analogous formula for conjugate transpose is $(AB{)}^{†}=B^{†}{A}^{†}$. Commutativity of scalar entries permits the last rearrangement; the matrix products themselves are not assumed to commute.

Over a field of characteristic two, the linear map $T:M_n(F)\to M_n(F)$, $A\mapsto A+A^T$, has kernel of a linear map the symmetric matrices, of dimension $n(n+1)/2$. Its image of a linear map is the linear subspace of symmetric matrices with zero diagonal, of dimension $n(n-1)/2$. Indeed, $T(A)=0$ means $A=A^T$, and every symmetric zero-diagonal matrix is the image of the matrix obtained by retaining just its entries above the diagonal. Thus the two dimensions still add to $n^2$, as required by the rank-nullity theorem.

For a matrix over the real numbers or complex numbers, applying the transpose twice to $A=\lambda A^T$ gives $A={\lambda }^{2}A$. A nonzero matrix is therefore possible only for $\lambda =1$ or $\lambda =-1$. These cases are precisely the symmetric matrices and skew-symmetric matrices, respectively. For every other scalar, the only solution is the zero matrix.