On the eigenvalues of Jordan products
This paper studies the possible eigenvalues of a Jordan product \(XA + AX\) for a given \(n\) by \(n\) matrix \(A\) and varying \(X\). A matrix \(A\) is called spectrally complete, if for any given \(n\) numbers \(c_i\) there exists a nonsingular matrix \(X\) so that \(XA+AX\) has the given eigenvalues \(c_i\). If the field \(F\) is algebraically closed and has characteristic different from 2 and if \(A\) is not a multiple of the identity \(I\), then it is shown that the \(n\) by \(n\) matrix \(A\) is spectrally complete if and only if \(n \leq 2\) rank\((A)\) and the minimal polynomial of \(A\) differs from \(x^2\).
scientific article; zbMATH DE number 1886840