Isometric Transformation Proof Using Matrix Form

[matt223]

Homework Statement

The questions asks for a proof that if a geometrical transformation R on a three dimensional vector space with metric \eta is length preserving, then R^{T}\eta R=\eta. Note that the summation convention is used throughout.

The transformation is given by
x'_{i}=R_{ij}x_{j}

Homework Equations

The length of a vector is determined by the metric according to
l^{2}=\eta _{ij}x_{i}x_{j}

The Attempt at a Solution

If R is length preserving then
l^{2}=\eta _{ij}x_{i}x_{j} =\eta _{ij}x'_{i}x'_{j}
and so
\eta _{ij}x_{i}x_{j}=\eta _{ij}R_{ip}x_{p}R_{jq}x_{q}

My question is how do I get from this stage to the desired relationship R^{T}\eta R=\eta. Perhaps this is already implied by the line above? If so, how?

PS: This is my first post here - thank you for any help!

 

[fzero]

Rearrange the Rs a bit in your expression:

<br /> \eta _{ij}x_{i}x_{j}=\eta _{ij}R_{ip}x_{p}R_{jq}x_{q} = (R_{ip}\eta _{ij}R_{jq})x_{p}x_{q}<br />

and try to rewrite R_{ip}\eta _{ij}R_{jq} in matrix form.