Question

Can I say that all orthonormal matrices are symmetric?

Answer 1

symmetric
\[A^T=A\] orthomormal
\[A^T=A^{-1}\] so no, they are different conditions

Answer 2

Somehow I seem to get almost all othornormal matrices as symmetric ones, which made me think like all othornormal matrices are symmetric.

Answer 3

\[\left[\begin{matrix}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]\]

is an orthonormal matrix that is not symmetric

Answer 4

oh wow.... so indeed there are orthonormal matrices that aren't symmetric. thanks for your help! :)

Answer 5

\[\left[\begin{matrix}\cos(\theta) & -\sin(\theta) \\ \sin(\theta)& \cos(\theta)\end{matrix}\right]\]
here is the rotation matrix about the origin for \[R^2\]

it is an orthonormal matrix that is not symmetric for most values of theta.

Answer 6

oh ya.... a rotation matrix is always an othornormal matrix, is this right?

Answer 7

yes

Answer 8

thanks a lot Zarkon! thank you so much!

Answer 9

np