Explain why the dot product of x and y equals the dot product of Px and Py. Then from (Px)^T (Py) = x^T y deduce that P^T P = I for any permutation. With x = (1,2,3) and y = (1,4,2) choose P to show that Px • y is not always x • Py.

Question

Explain why the dot product of x and y equals the dot product of Px and Py. Then from (Px)^{T} (Py) = {x}^{T}y deduce that {P}^{T}P = I for any permutation. With x = (1,2,3) and y = (1,4,2) choose P to show that Px • y is not always x • Py.

Accepted Answer

[latex](Px)^{T} (Py) = {x}^{T}{P}^{T}Py ={x}^{T}y[/latex] since [latex]{P}^{T}P = I[/latex].

In general

[latex]Px • y=x • {P}^{T}y eq x • Py[/latex]:

Non-equality where [latex]P eq {P}^{T}[/latex]:

[latex]\left[ \begin{matrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0 \end{matrix} ight] \left[ \begin{matrix} 1 \ 2 \ 3 \end{matrix} ight] \cdot \left[ \begin{matrix} 1 \ 1 \ 2 \end{matrix} ight] eq \left[ \begin{matrix} 1 \ 2 \ 3 \end{matrix} ight] \cdot \left[ \begin{matrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0 \end{matrix} ight] \left[ \begin{matrix} 1 \ 1 \ 2 \end{matrix} ight][/latex].

Question 2.7.12: Explain why the dot product of x and y equals the dot produc...

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