The product of two quadratic matrices (Q5943024)

Let \(K\) be a field and let \(p(x):=(x-\beta)(x-\beta^{-1})\in K[x]\) where \(\beta^{2}\neq\beta^{-2}\). Consider the algebra \(K^{n\times n}\) of \(n\times n\) matrices over \(K\). The authors call a matrix \(M\in K^{n\times n}\) ``quadratic'' (for \(p)\) if \(p(M)=0\), and call it ``\(2\)-quadratic'' if it is the product of two quadratic matrices. The main theorem characterizes \(2\)-quadratic matrices (Theorem 1.2): \(M\) is \(2\)-quadratic if and only if it is similar to a block diagonal matrix of the form \(\operatorname {diag}(X,Y,\beta ^{2}I,\beta^{-2}I)\) (some blocks may be missing) where: (1) \(X\) is similar to \(X^{-1}\) and either \(\text{char}(K)=2\) or \(X\) has no elementary divisor of the form \((x+1)^{t}\) with \(t\) odd; and (2) \(Y= \operatorname {diag}(\beta^{2}I+ST,(\beta^{-2} I+TS)^{-1})\) where \(S\in K^{r\times s}\) and \(T\in K^{s\times r}\) and both \(TS\) and \(ST\) are nilpotent. A critical step in the proof is the following proposition (Prop. A.1): If \(X\in K^{2m\times 2m}\) satisfies condition (1) above, then \(X\) is similar to a matrix of the form \(\left[\begin{smallmatrix} I & B\\ A & I+AB \end{smallmatrix}\right] \) for some \(A,B\in K^{m\times m}.\)