Exercise 2.7.12

Answers

We have

{(P x)}^{T} (P y) = x^{T} P^{T} P y = x^{T} y

since $P^{T} P = I$ . In general,

P x \cdot y = x \cdot P^{T} y \neq x \cdot P y

Non-equality where $P \neq P^{T}$ :

[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] \cdot [\begin{matrix} 1 \\ 1 \\ 2 \end{matrix}] \neq [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] \cdot [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}] [\begin{matrix} 1 \\ 1 \\ 2 \end{matrix}] .

mathchakra

2022-01-16 12:18