Vectors as special matrices

2.7.1 Vectors as special matrices

Vectors are matrices, too

\vec{a} = (\begin{matrix} a_{11} \\ \dots \\ a_{M 1} \end{matrix}) \Rightarrow M \times 1 matrix! column vector!

transposed {\vec{a}}^{⊤} = (a_{11}, \dots, a_{M 1}) \Rightarrow 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: $\underset{M \times P}{\underset{⏟}{\tilde{C}}} = \tilde{A} \cdot \tilde{B}$ is defined as the product of $\tilde{A}$ and $\tilde{B}$ with:

C_{l s} = \sum_{j = 1}^{N} a_{l j} b_{j s} \begin{array}{l} l = 1, \dots, M \\ s = 1, \dots, P \end{array}} rule: line \times column!

Examples:

\begin{array}{cccc} \overset{M}{2} \times \overset{N}{3} & \overset{N}{3} \times \overset{P}{2} & \binom{l’th line of \tilde{A}}{s’th row of \tilde{B}}} & c_{l s} = 2 \times 2 \\ (\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}) & (\begin{array}{cc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}) & = & (\begin{array}{cc} 14 & 32 \\ 32 & 77 \end{array}) \end{array}

With frame

Matrices