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
\[\sum _ {i=1}^n \pmb a _ i^\intercal \pmb a _ i\]
?
\[A^\intercal A\]