A necessary condition for quantitative exponential stability of linear state-space systems

For an arbitrary n × n constant matrix A the two following facts are well known: • (1/n)Re(trace A) - maxj=_i,...,nRe λ_j(A4)≤0; • If U is a unitary matrix, one can always find a skew-Hermitian matrix A so that U = e^A. In this note we present the extension of these two facts to the context of linear time-varying dynamical systems ẋ = A(t)x, t≥0. As a by-product, this result suggests that, the notion of "slowly varying state-space systems", commonly used in literature, is mathematically not natural to the problem of exponential stability.