With frame
Vectors are matrices, too
| \[\vec{a}=\vect{a_{11}\\\cdots\\a_{M1}}\Rightarrow \mbox{$M\times1$ matrix! column vector!}\] |
| \[\mbox{transposed}\quad \vec{a}^\top=\left(a_{11},\ldots,a_{M1}\right)\Rightarrow \mbox{$1\times M$ matrix! row or line vector!}\] |
Product of two matrices:
Definition 23 \(\tilde A\) an \(M\times N\) matrix and \(\tilde B\) is an \(N\times P\) matrix then: \(\underbrace{\tilde C}_{M\times P}=\tilde{A}\cdot\tilde{B}\) is defined as the product of \(\tilde{A}\) and \(\tilde B\) with:
| \[C_{ls}=\sum_{j=1}^N a_{lj}\,b_{js}\;\left.\begin{array}{l}l=1,\ldots,M\\s=1,\ldots,P\end{array}\right\}\mbox{{rule:} line $\times$ column!}\] |
Examples:
| \[\begin{array}{cccc} \stackrel{M}{2}\times\stackrel{N}{3} & \stackrel{N}{3}\times\stackrel{P}{2} & \left. \genfrac{}{}{0pt}{}{\mbox{l'th line of $\tilde A$}}{\mbox{s'th row of $\tilde B$}}\right\}& c_{ls}=2\times2\\ \left(\begin{array}{ccc}1&2&3\\4&5&6\end{array}\right) & \left(\begin{array}{cc}1&4\\2&5\\3&6\end{array}\right) & = & \left(\begin{array}{cc}14&32\\32&77\end{array}\right)\end{array}\] |
© J. Carstensen (Math for MS)
Matrices