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?
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
2 Asked on November 6, 2021 by navyism
1 Asked on November 6, 2021
adjoint operators banach spaces functional analysis operator theory
0 Asked on November 6, 2021 by marcelo-rm
2 Asked on November 6, 2021 by exodd
examples counterexamples trigonometric series trigonometry upper lower bounds
1 Asked on November 6, 2021
calculus convergence divergence power series sequences and series
1 Asked on November 6, 2021 by plasmacel
algebraic geometry euclidean geometry geometry linear algebra vectors
3 Asked on November 6, 2021 by jacobcheverie
elementary set theory functions proof verification proof writing
1 Asked on November 6, 2021
3 Asked on November 6, 2021 by domonoxyledyl
2 Asked on November 6, 2021 by josef-hlava
2 Asked on November 6, 2021 by dhrubajyoti-bhattacharjee
2 Asked on November 6, 2021 by aladin
4 Asked on November 6, 2021
5 Asked on November 6, 2021
3 Asked on November 6, 2021 by charlie-chang
1 Asked on November 6, 2021
algebraic geometry commutative algebra derived functors schemes sheaf cohomology
1 Asked on November 6, 2021 by pythusiast
Get help from others!
Recent Answers
Recent Questions
© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP