Slices of Matrices — a Scenario for Spectral Theory

Given a real, symmetric matrix S, we define the slice FS through S as being the connected component containing S of two orbits under conjugation: the first by the orthogonal group, and the second by the upper triangular group. We describe some classical constructions in eigenvalue computations and integrable systems which keep slices invariant — their properties are clarified by the concept. We also parametrize the closure of a slice in terms of a convex polytope. The basic definition Let S be a real n × n symmetric matrix with simple spectrum σ(S) = {λ1 > . . . > λn}. When is a matrix simultaneously an orthogonal and an upper triangular conjugation of S? More precisely, we consider the slice FS through S, defined to be the intersection of the sets {QSQ, for arbitrary real orthogonal matrices Q with detQ = 1} and {RSR, for arbitrary real upper triangular matrices R with diagR > 0}. How large is a slice? Clearly, if QSQ = RSR then SQR = QRS, and hence, since S has simple spectrum, QR = f(S), for some (real) polynomial f . The matrices Q and R are uniquely determined from f(S) if f(S) is invertible: this is the standard QR-factorization of a matrix ([7]). For convenience, we write f(S) = QR = [