Factor Matrix in the Inner Product

Theorem

Let $\mathbf u$ and $\mathbf v$ be $1 \times n$ column vectors.

Then:

$\innerprod {A \mathbf u} {\mathbf v} = \innerprod {\mathbf u} {A^\intercal \mathbf v}$

Proof

\(\ds \innerprod {A \mathbf u} {\mathbf v}\)

\(=\)

\(\ds \paren {A \mathbf u}^\intercal \mathbf v\)

Definition of Dot Product

\(\ds \)

\(=\)

\(\ds \mathbf u^\intercal A^\intercal \mathbf v\)

Transpose of Matrix Product

\(\ds \)

\(=\)

\(\ds \innerprod {\mathbf u} {A^\intercal \mathbf v}\)

Definition of Dot Product

$\blacksquare$