• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Invariance of quadratic form for orthogonal matrices

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?
 

PeroK

Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
9,112
3,284
Looks good to me.
 
Thanks!
 

Want to reply to this thread?

"Invariance of quadratic form for orthogonal matrices" You must log in or register to reply here.

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
Top