Matrix Product as a Sum of Outer Product
Table of contents
1. Outer Product : rank one matrix
\[\mathbf{u} \bigotimes_{outer} \mathbf{v} = \mathbf{u} \times \mathbf{v} = \mathbf{u} \mathbf{v}^T \text{ : rank one matrix}\]row space of $ \mathbf{u} \mathbf{v}^T = C((\mathbf{u} \mathbf{v}^T)^T) \text{ : the line through } \mathbf{v}$
cf) inner product : $ \mathbf{u} \cdot \mathbf{v} = <\mathbf{u}, \mathbf{v}> = \mathbf{u}^T \mathbf{v} \text{ : scalar}$
cf) cross product : $ \mathbf{u} \times \mathbf{v} = \mathbf{u} \bigotimes_{cross} \mathbf{v} \text{ : vector} $
2. Matrix Product as a Sum of Outer Product
\[AB = \begin{pmatrix} \mathbf{a}_{1} & \cdots & \mathbf{a}_{n} \end{pmatrix} \begin{pmatrix} \mathbf{b}_1^* \\ \vdots \\ \mathbf{b}_n^* \end{pmatrix} = \mathbf{a_1}\mathbf{b_1^*} + \mathbf{a_2}\mathbf{b_2^*} + \cdots + \mathbf{a_n}\mathbf{b_n^*} = \sum_i^n \mathbf{a_i}\mathbf{b_i^*}\]3. total counts
\[AB = (m \text{ by } n) \text{ times } (n \text{ by } p)\]- inner product : mp inner product, n multiplications each : mnp
- outer product : n outer product, mp multiplications each : mnp