Orthogonal matrices $A$ for which $A^n = I$

Question

Can we say that orthogonal matrices for which $P^n = I$ are necessarily permutation matrices?

I know the matrices that satisfy the condition $P^n = I$ are called periodic. Also, permutation matrices are non-negative and orthogonal. Is the condition of non-negativity inherent in periodic matrices?

If not, Could we put restrictions on periodic matrices for them to be permutations? Or on orthogonal periodic matrices for them to be permutations?

Any help without using the entries of the matrix will be appreciated.

I do not know much about properties of periodic matrices. It'll be a great help if someone gives better insight of such matrices.

Answer 1

A very trivial observation is that, let $E_{ij}$ be the matrix with 1 at the $i$-th row and $j$-th column, and 0 elsewhere. Then $E_{1i}PE_{j1}$ is just the matrix with $P_{ij}$ at the upper left corner, and 0 elsewhere. If, additionally, you allow the use of absolute values and determinants, you can construct a huge expression, basically saying "($P_{11}=1$ and $P_{22}=1$ and ...) or ($P_{21}=1$ and $P_{12}=1$ and $P_{33}=1$ and ...)", which fulfills your criterion.

Answer 2

No. These matrices are roots of the unit matrix, not permutations. Counter example : $ A={1\over\sqrt 2}\begin{pmatrix}1&1\\-1&1 \end{pmatrix}$

$A^4=I$

Answer 3

$${\bf A} := \begin{bmatrix}\cos{\left(\frac{2 \pi}{n} \right)} & - \sin{\left(\frac{2 \pi}{n} \right)}\\\sin{\left(\frac{2 \pi}{n} \right)} & \cos{\left(\frac{2 \pi}{n} \right)}\end{bmatrix}$$

is orthogonal and ${\bf A}^n = {\bf I}_2$. However, $\bf A$ is not a permutation matrix.