ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4591 on D398: Matrix transpose

Binary additivity of transpose for real matrices

Formulation 0

Let $A, B \in \mathbb{R}^{N \times M}$ each be a D4571: Real matrix.

Then \begin{equation} \left( A + B \right)^T = A^T + B^T \end{equation}

Proofs

Proof 0

Let $A, B \in \mathbb{R}^{N \times M}$ each be a D4571: Real matrix.

Fix $n \in \{ 1, \ldots, N \}$ and $m \in \{ 1, \ldots, M \}$. Then \begin{equation} (A + B)^T_{n, m} = (A + B)_{m, n} = A_{m, n} + B_{m, n} = A^T_{n, m} + B^T_{n, m} \end{equation} $\square$