Products of two involutions with prescribed eigenvalues and some applications Original Research Article

Let F be a field. In [Djokovic, Product of two involutions, Arch. Math. 18 (1967) 582–584] it was proved that a matrix Aset membership, variantFn×n can be written as A=BC, for some involutions B,Cset membership, variantFn×n, if and only if A is similar to A-1. In this paper we describe the possible eigenvalues of the matrices B and C. As a consequence, in case charF≠2, we describe the possible similarity classes of (P11circled plusP22)P-1, when the nonsingular matrix P=[Pij]set membership, variantFn×n, i,jset membership, variant{1,2} and P11set membership, variantFs×s, varies. When F is an algebraically closed field and charF≠2, we also describe the possible similarity classes of [Aij]set membership, variantFn×n, i,jset membership, variant{1,2}, when A11 and A22 are square zero matrices and A12 and A21 vary.