Products of two involutions with prescribed eigenvalues and some applications
Let \(A\) be an \(n\times n\) invertible matrix over a field \(F\). Then \textit{D. Ž. Djoković} [Arch. Math. 18, 582--584 (1967; Zbl 0153.35502)] proved that \(A\) is a product \(BC\) of two involutions (matrices whose square is \(I\)) if and only if \(A\) is similar to \(A^{-1}\). The main theorem of the present paper is the following. If \(\text{char}(F)\neq2\) then the involutions \(B\) and \(C\) can be chosen to have, respectively, \(r\) and \(s\) eigenvalues equal to \(1\) if and only if: (i) \(\det(A)=(-1)^{r+s}\); and (ii) \(\left| s+r-n\right| \leq k_{1}\) and \(\left| s-r\right| \leq k_{2}\) where \(k_{1}\) and \(k_{2}\) are the number of elementary divisors of \(A\) of odd degree associated with the eigenvalues \(1\) and \(-1\), respectively. Using similar methods the author also proves related results such as the following theorem. Suppose that \(C\) is an \(n\times n\) matrix over a field \(F\) where \(F\) is algebraically closed with \(\text{char}(F)\neq2.\) Then \(C\) is similar to a matrix of the form \[ \left[ \begin{matrix} 0 & X\\ Y^{\top} & 0 \end{matrix} \right] \text{ where }X\text{ and }Y\text{ are }s\times(n-s)\text{ matrices} \] if and only if (i) \(C\) and \(-C\) are similar; and (ii) \(\left| 2s-n\right| \leq k\) where \(k\) is the number of elementary divisors of \(C\) of odd degree associated with the eigenvalue \(0\).
scientific article; zbMATH DE number 5349140
scientific article