# Behaviour of orthogonal matrices

Mathematics Asked by a9302c on December 15, 2020

I am given that A is an orthogonal matrix of order $$n$$, and $$u, v$$ are Vectors in the $$R^n$$ space.

I need to prove that $$||u|| = ||Au||$$. The first step of the solution hint I am given is that $$||Au||^2 = (Au)^T(Au)$$. Why is this so? I know that $$A^{-1} = A^T$$ in the definition of an orthogonal matrix, but how does this contribute to the above statement? Or is there some other property I’m missing out on?

## 2 Answers

Assume that $$A$$ is orthogonal, i.e. that $$A^T A = I$$ (observe that this is equivalent to $$A^{-1} = A^T$$). We consider

$$|| A x ||^2 = (Ax)^T (Ax) = x^T A^T A x = x^T x = || x || ^2,$$

which shows that $$|| A x || = || x ||$$.

Comment: On advise from another user, I posted a more thorough version of my earlier comment as this answer.

Correct answer by Fenris on December 15, 2020

If $$u=(u_1,u_2,ldots,u_n)$$, thenbegin{align}u^top u&=begin{pmatrix}u_1\u_2\vdots\u_nend{pmatrix}(u_1,u_2,ldots,u_n)\&=u_1^{,2}+u_2^{,2}+cdots+u_n^{,2}\&=|u|^2.end{align}

Answered by José Carlos Santos on December 15, 2020

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