Factor Matrix in the Inner Product

Theorem

$\left \langle {A \mathbf u,\mathbf v} \right \rangle = \left \langle {\mathbf u, A^T\mathbf v} \right \rangle$

where $\mathbf u$ and $\mathbf v$ are both $1 \times n$ column vectors.

Proof

By definition, the meaning of the above notation is $\left \langle {\mathbf u, \mathbf v} \right \rangle = \mathbf u^T \mathbf v$, so we get:

$\left \langle {A \mathbf u,\mathbf v} \right \rangle = \left({A \mathbf u}\right)^T \mathbf v$

From Transpose of Matrix Product, we have $(AB)^T = B^T A^T$, and so we get:

$\left \langle {A \mathbf u, \mathbf v} \right \rangle = \mathbf u^T A^T \mathbf v$

But this is, by definition, the same as $\left \langle {\mathbf u, A^T \mathbf v} \right \rangle$.

$\blacksquare$