# How to find this rotation?

Let $\{e_1,e_2\}$ be standard orthonormal basis of $\Bbb R^2$. Then a new basis $\{E_1,E_2\}$ obtained by $\frac{\pi}{4}$-rotation (counterclockwise) of $\{e_1,e_2\}$ is:

$$R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}\Rightarrow (E_1,E_2)= \frac{\sqrt{2}}{2}\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ \end{bmatrix}.$$ For example in dim $=3$ the rotation of standard basis are: first rotation in $xy$-plan about $z$-axis and second rotation in direction of old $xz$-plan (before rotation)

I want to know

Question: How to find matrix of $\frac{\pi}{4}$-rotation ($\frac{\pi}{4}$ about every coordinate axis) for $\Bbb R^n$? Does this matrix belongs to ${\rm SO}(n)$ or ${\rm O}(n)$?

• In $\Bbb R^n$, a rotation is not about an axis, it's in a plane. In $\Bbb R^3$, that's the same thing, but in $\Bbb R^4$, rotations aren't about an axis, they're about a plane (unless you want to call that plane the "axis plane", which seems a little strange to me). – Arthur Sep 8 '17 at 11:59
• What do you mean for a rotation about every coordinate axis. That doesn't seem possible (if you mean a rotation all at once). – Michael Burr Sep 8 '17 at 12:00
• for example in dim $=3$ the rotation of standard basis are: first rotation in $xy$-plan about $z$-axis and second rotation in direction of old $xz$-plan (before rotation) – C.F.G Sep 8 '17 at 12:06
• – Widawensen Sep 8 '17 at 13:04
• The discussion here may be of help to you. – Epiousios Sep 16 '17 at 21:05

Any rotation in $\mathbb{R}^n$ can be represented by a matrix with unit determinant. Also, you can combine any number of rotations by multiplying the matrices they are represented by.
Rotations in $n$ dimensions are represented by a matrix of order $n$, of course.
You could think of it this way: in $\mathbb{R}^3$ you can construct a matrix to rotate any vector along the xy-plane just by placing that matrix you used for $\mathbb{R}^2$ and adding a $[0, 0]^T$ vector to the right, and a $[0, 0, 1]$ to the bottom. You can think of every $n$-th coloumn/row pair as referring to each possible axis.
Carefully placing $\sin$ and $\cos$ in a matrix as you've shown, and leaving 1's on the diagonal, gives you a rotation along a certain plane in $n$ dimensions. And you get the idea that multiplying each matrix will get you the rotation you're looking for.