Isospectral Deformations of Diagonal Matrices by Symmetric Matrices in Dimension Four

Abstract

For a diagonal matrix $D$ with distinct diagonal entries in an algebraically closed field $K$ we study the symmetric matrices $Q$ with entries in $K$ such that the affine line $D+tQ$ has the same eigenvalues in $K$ as $D$. We study the equality of characteristic polynomials, $χ_{D+tQ}(x)=χ_D(x) \text{ for all }\: t\in K$ and examine the resulting system of equations. For generic diagonal matrices of size at most $4$ we show that only the trivial symmetric matrix $Q=0$ satisfies this isospectral condition. We use this result to compute the degree of a field extension that appears in the invariant theory of quadratic forms under congruence action.