Matrix Transpose

Let $A\in M_{m\times n}(\mathbb{R})$. The transpose of $A$, denoted by $A^T$, is the $n\times m$ matrix satisfying $(A^T{)}_{ij}=(A{)}_{ji}$.

Properties of Transpose. Let $A,B\in M_{m\times n}(\mathbb{R})$ and $c\in \mathbb{R}$. Then $A^T\in M_{m\times n}(\mathbb{R})$ $(A^T{)}^T=A$ $(A+B{)}^T=A^T+B^T$ $(cA{)}^T=c{A}^T$

A matrix $A$ is symmetric if $A^T=A$.

Other property: $(AB{)}^T=B^T{A}^T$