What is the line of reflection of this 3x3 matrix?

Question

I'm having a difficulty in determining the line of reflection of the following matrix:

\begin{bmatrix}1&0&0\\0&\frac12&\frac {\sqrt{3}}2\\0&\frac {\sqrt{3}}2&-\frac 12\end{bmatrix}

I can see that the x-component of a 3-dimensional vector remains unchanged after the linear transformation. I also noticed that the lower right block is a reflection matrix.

I would be able to find the line of reflection of a 2x2 matrix, but I have no idea how to take account of the x-component in a 3x3 matrix. How do I describe this matrix geometrically?

Answer 1

In this case, it'll be a plane of reflection. If you consider restricting the transformation to just the $yz$-plane and subsequently work with just the lower right hand $2 \times 2$ matrix, you can find the line of reflection in the $yz$-plane (as you would with any ordinary $2 \times 2$ matrix). The plane of reflection for the whole transformation in $\mathbb{R}^3$ will be this line together with the $x$-coordinate allowed to vary freely (i.e. the plane orthogonal to the $yz$-plane that cuts through that line).