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?

norm orthogonal matrices transpose vector spaces

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

Behaviour of orthogonal matrices

2 Answers

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