Notes - Continuous Mathematics HT23, Misc
Flashcards
If
\[\mathbf{v}=\left[\begin{array}{c}
v_{1} \\\\
v_{2} \\\\
\vdots \\\\
v_{n}
\end{array}\right], \mathbf{w}=\left[\begin{array}{c}
w_{1} \\\\
w_{2} \\\\
\vdots \\\\
w_{m}
\end{array}\right]\]
then what is the outer product $\pmb v \otimes \pmb w$?
\[\pmb v \pmb w^\intercal =\left[\begin{array}{cccc}
v_{1} w_{1} & v_{1} w_{2} & \cdots & v_{1} w_{m} \\\\
v_{2} w_{1} & v_{2} w_{2} & \cdots & v_{2} w_{m} \\\\
\vdots & \vdots & \ddots & \vdots \\\\
v_{n} w_{1} & v_{n} w_{2} & \cdots & v_{n} w_{m}
\end{array}\right]\]
If
\[\mathbf{v}=\left[\begin{array}{c}
v_{1} \\\\
v_{2} \\\\
\vdots \\\\
v_{n}
\end{array}\right], \mathbf{w}=\left[\begin{array}{c}
w_{1} \\\\
w_{2} \\\\
\vdots \\\\
w_{m}
\end{array}\right]\]
then what is the $(i, j)$th entry in the outer product $\mathbf v \otimes \mathbf w$?
\[(\mathbf v \otimes \mathbf w)_{ij} = v_i w_j\]
Given an $n \times m$ matrix $A$ with rows $\pmb a _ i$, how can you simplify
\[\sum_{i=1}^n \pmb a_i^\intercal \pmb a_i\]
?
\[A^\intercal A\]