# 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?

Related Calculus and Beyond Homework News on Phys.org

#### PeroK

Homework Helper
Gold Member
2018 Award
Looks good to me.

#### spaghetti3451

Thanks!

"Invariance of quadratic form for orthogonal matrices"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving