# Invariance of quadratic form for orthogonal matrices

#### spaghetti3451

1. The problem statement, all variables and given/known data

Show that all $n \times n$ (real) orthogonal matrices $O$ leave invariant the quadratic form $x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}$, that is, that if $x'=Ox$, then $x'^{2}=x^{2}$.

2. Relevant equations

3. The attempt at a solution

$x'^{2} = (x')^{T}(x') = (Ox)^{T}(Ox) = x^{T}O^{T}Ox = x^{T}x = x^{2}$.

I would like to check if I am correct?

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#### PeroK

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Looks good to me.

Thanks!

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