Transpose

For an $m\times n$ matrix $A$, we define the transpose as an $n\times m$ matrix $A^T$, where the elements in the rows and columns are swapped. For vectors, the transpose simply switches its type between a row and column vector (to the one it is not).

If $A=A^T$, then $A$ is symmetric.

In MATLAB, we can use transpose(A). In NumPy, we use np.transpose(A).

Properties

$$(A^T{)}^T=A$$ $$(A+B{)}^T=A^T+B^T$$ $$(kA{)}^T=k{A}^T$$ $$(AB{)}^T=B^T{A}^T$$ $$(A^T{)}^{-1}=(A^{-1}{)}^T$$ $$\text{rank}(A)=\text{rank}(A^T)$$ $$\text{det}(A)=\text{det}(A^T)$$