Invariance of quadratic form for orthogonal matrices

[spaghetti3451]

Homework Statement

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 $x'^{2}=x^{2}$.

Homework Equations

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?

 

[PeroK]

Looks good to me.

 

[spaghetti3451]

Thanks!