book:chap2:2.7_the_inner_product
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| book:chap2:2.7_the_inner_product [2022/04/01 21:01] – jv | book:chap2:2.7_the_inner_product [2022/04/22 23:46] (current) – jv | ||
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| b) linearity in the first argument: | b) linearity in the first argument: | ||
| - | $\langle c \, \mathbf v \mid \mathbf w \rangle = c \; \langle \mathbf v \mid \mathbf w \rangle$ \\ | + | $\langle c \odot \mathbf v \mid \mathbf w \rangle = c \; \langle \mathbf v \mid \mathbf w \rangle$ \\ |
| - | and $\langle \mathbf u + \mathbf v \mid \mathbf w \rangle = \langle \mathbf u \mid \mathbf w \rangle + \langle \mathbf v \mid \mathbf w \rangle$\\ | + | and $\langle \mathbf u \oplus |
| c) positivity: | c) positivity: | ||
| Line 58: | Line 58: | ||
| Conjugate symmetry and linearity for the first argument imply the following relations for the second argument | Conjugate symmetry and linearity for the first argument imply the following relations for the second argument | ||
| \begin{align*} | \begin{align*} | ||
| - | \langle \mathbf v \mid c \, \mathbf w \rangle | + | \langle \mathbf v \mid c \odot \mathbf w \rangle |
| - | &= \overline{\langle c \, \mathbf w \mid \mathbf v \rangle} | + | &= \overline{\langle c \odot \mathbf w \mid \mathbf v \rangle} |
| = \bar{c} \; | = \bar{c} \; | ||
| = \bar{c} \; \langle \mathbf v \mid \mathbf w \rangle | = \bar{c} \; \langle \mathbf v \mid \mathbf w \rangle | ||
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \langle \mathbf u \mid \mathbf v + \, \mathbf w \rangle | + | \langle \mathbf u \mid \mathbf v \oplus \mathbf w \rangle |
| - | &= \overline{\langle \mathbf v + \mathbf w \mid \mathbf u \rangle} | + | &= \overline{\langle \mathbf v \oplus |
| = \overline{\langle \mathbf v \mid \mathbf u \rangle} + \overline{\langle \mathbf w \mid \mathbf u \rangle} | = \overline{\langle \mathbf v \mid \mathbf u \rangle} + \overline{\langle \mathbf w \mid \mathbf u \rangle} | ||
| = \langle \mathbf u \mid \mathbf v \rangle + \langle \mathbf u \mid \mathbf w \rangle | = \langle \mathbf u \mid \mathbf v \rangle + \langle \mathbf u \mid \mathbf w \rangle | ||
| \end{align*} | \end{align*} | ||
| - | |||
| </ | </ | ||
book/chap2/2.7_the_inner_product.1648839686.txt.gz · Last modified: 2022/04/01 21:01 by jv