On products of involutions

Question

Is it possible to write $$M = \begin{bmatrix}x&0\\0&-\dfrac{1}{x}\end{bmatrix}, x \in \mathbb{R} \backslash \{0\}$$ into a product of two involutory real matrices? I know that it can be written as a product of three but is it possible to reduce it to two?

Answer 1

$M$ can be written as product of two involutions if and only if $x=\pm 1$.

To see this, let us first observe the following : If $M=M_1M_2$ such that $M_i^2=1$ for $i=1,2$, then $M_1MM_1^{-1}=M_1^2M_2M_1^{-1}=M_2^{-1}M_1^{-1}=M^{-1}$. So if possible let there exist such a matrix $M_1=\begin{bmatrix}a&b\\c&d \end{bmatrix}$. Then $M_1M=M^{-1}M_1$ implies $b=c=0$ and $a(x^2-1)=d(x^2-1)=0$. Therefore if $x\neq \pm 1$, then $a=d=0$.