On the invariant polynomials of Jordan products
Consider the ring \(F^{n\times n}\) of all \(n\times n\) matrices over an algebraically closed field \(F\) of characteristic \(\neq 2\). For each \(A\in F^{n\times n}\) define \(R_{F}(A)\) to be the minimum of \text{rank}\((A-\lambda I)\) for \(\lambda\in F\). It is known [see \textit{G. N. Oliveira, E. M. Sá} and \textit{J. A. Dias de Silva}, Linear Multilinear Algebra 5, 119-128 (1977; Zbl 0367.15010)] that \(n-R_{F}(A)\) is equal to the number of nonconstant invariant factors of \(A\). In the present paper the authors prove that, for each \(A\in F^{n\times n}\), the value of \(R_{F}(XA+AX)\) runs over \(0,1,\dots ,2 \text{rank}(A)\) as \(X\) runs over \(F^{n\times n}\). They also prove a similar, but more complicated, theorem (involving the degrees of the invariant factors of \(A)\) for the values of \(R_{F}(XA+AX)\) when \(X\) is restricted to the invertible matrices in \(F^{n\times n}\).
scientific article; zbMATH DE number 1880033