# 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

## Related Questions

### A variant version of Euler’s phi function

2  Asked on November 6, 2021 by navyism

### If $Tcolon X to Y$ is such that $T^*colon Y^* to Y^*$, what does this imply about $T$?

1  Asked on November 6, 2021

### How do I prove a half open interval is a sigma algebra?

0  Asked on November 6, 2021 by marcelo-rm

### Bound on truncated trigonometric polynomial

2  Asked on November 6, 2021 by exodd

### Parent and childs of a full d-node tree

3  Asked on November 6, 2021 by aggelos

### For which values of x does the power series converge or diverge?

1  Asked on November 6, 2021

### Intersection of lines in higher dimensions

1  Asked on November 6, 2021 by plasmacel

### Finite Sets, Equal Cardinality, Injective $iff$ Surjective.

3  Asked on November 6, 2021 by jacobcheverie

### Pair of tangent lines from a point $(x_1,y_1)$ to a circle $x^2+y^2=a^2$

1  Asked on November 6, 2021

### How to define a curve surrounding the origin?

3  Asked on November 6, 2021 by domonoxyledyl

### What would a Cayley table of inverse semigroup look like?

2  Asked on November 6, 2021 by josef-hlava

### about prisoners and selection of numbers

1  Asked on November 6, 2021

### If a sequence $langle a_nrangle$ is such that $a_1a_2=1, a_2a_3=2, ldots$ and $limfrac{a_n}{a_{n+1}}=1.$ Then find $|a_1|.$

2  Asked on November 6, 2021 by dhrubajyoti-bhattacharjee

### $F= bigcap_{i=1}^{infty} F_i$ isn’t necessarily connected where $F_{i+1} subseteq F_i$ and $F_i subseteq mathbb{R}^2$ are closed and connected

2  Asked on November 6, 2021 by aladin

### plot of $sin(x) + sin(y)= cos(x) + cos(y)$

4  Asked on November 6, 2021

### Is it possible to show that the fifth roots of 1 add up to 0 simply by using trigonometric identities?

5  Asked on November 6, 2021

### Convergence of $sumlimits_{n=1}^inftyleft{frac{1cdot 3dots 2n-1 }{2cdot 4dots 2n}cdotfrac{4n+3}{2n+2}right}^2$

3  Asked on November 6, 2021 by charlie-chang

### Stalks of Higher direct images of structure sheaf at smooth points

1  Asked on November 6, 2021

### Convolution: Integral vs. Discrete sum

1  Asked on November 6, 2021 by pythusiast

### What can be concluded about an equilibrium point that has finitely many trajectories that converge to it?

0  Asked on November 6, 2021 by abc1455

### Ask a Question

Get help from others!

© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP