Eigenvectors and geometrical transformation

Question

$$A= \begin{pmatrix} 2/3 & 2/3 & -1/3 \\ 2/3 & -1/3 & 2/3 \\ -1/3 & 2/3 & 2/3 \\ \end{pmatrix}$$, I need to understand that kind of geometric trasformation this matrix represents. Characteristic polynomial is $-1 + λ + λ^2 - λ^3 = (1-λ^2)(λ-1)$, eigenvalues are $λ_{1,2,3}=1,1,-1$, eigenvectors $v_{1,2,3}=[(2,1,0)^T,(-1,0,1)^T,(1,-2,1)^T]$. Now I should be able to tell, around which vector $u$ matrix rotates vectors, and if there is a symmetry to plane, perpendicular to $u$. Solution says to me that this matrix makes a symmetry to plane, perpendicular to $(1,-2,1)^T$, my problem is what is have no idea, which criterium it uses to choose "right" vector as axis of rotation. Can someone please make these criteria obvious to me?

Answer 1

If you have eigenvector and eigenvalue pairs

\begin{align*} v_1 &= (2, 1, 0)^T, &\lambda_1 = 1 \\ v_2 &= (-1, 0, 1)^T, &\lambda_2 = 1 \\ v_3 &= (1, -2, 1)^T, &\lambda_3 = -1 \end{align*}

I would expect this to be a reflection, rather than a rotation. There are two reasons this would be so.

First, the determinant is $-1 = 1^2 \cdot (-1)$, the product of your eigenvalues. Not all transformations with determinant $-1$ are reflections, but all reflections have determinant $-1$. Rotations have determinant $1$. That's why you're having trouble finding an axis of rotation; there isn't one!

To think about this reflection geometrically, let $W = \operatorname{Span}\big(\{v_1, v_2\}\big)$. It's clear that your transformation $A$ leaves $W$ fixed (that is, we have A(W) = W). Further, it fixes every $w \in W$; we have $A(w) = w$.

Now $A$ fixes the hyperplane $W$ point-wise, what does it do to vectors perpendicular to $W$? It's not hard to see that $v_3$ is perpendicular to both $v_1$ and $v_2$, which span $W$, so we must have $v_3 \perp W$. Conveniently, $A(v_3) = -v_3$. This is exactly what reflections through a hyperplane containing the origin do: In an $n$-dimensional space, they fix every vector in a particular $(n-1)$-dimensional subspace $W$, while transforming vectors normal to $W$ to their negative.