Does there exist a non-symmetric involutory matrix?

Question

Let $A$ be a real involutory matrix i.e. $$A^2 = I.$$

Is it necessarily symmetric?

Any help will be highly appreciated. Thank you very much.

Answer 1

It is easy to construct 2-dimensional counterexamples: $A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$ is involutory if $a+d=0$ and $a^2+bc=1$. In particular, $A = \begin{pmatrix}0 & b \\ 1/b & 0\end{pmatrix}$ with $b \ne 0, 1$ is a non-symmetric, involutory matrix.

Answer 2

No; simply take $$A = \begin{pmatrix}\frac 12 & 3 \\ \frac 14 & -\frac 12\end{pmatrix}.$$

Answer 3

There is an insight that might be explained. You can produce infinite examples of this type reasoning on the fact that an involutory matrix is a symmetry and is a symmetric matrix if and only if it represents an orthogonal symmetry. It is sufficient tho choose two subspaces non mutually orthogonal such that thei direct sum gives back all the space and construct a symmetry on one of the two with direction the other to produce a counterexample.