Two-Sided Singular-Value Bounds and Separable Stability Metrics for Sylvester Matrix Differential Equations

We develop two-sided, order-free bounds for the Sylvester matrix differential equation X˙(t)=AX(t)+X(t)B+C,X(t0)=X0, allowing rectangular and nonsymmetric solutions. We derive upper and lower bounds for each singular value of X(t). A Dini-derivative argument yields a sharper lower forcing term than the one obtained from a direct global reverse-triangle estimate. Corresponding two-sided Frobenius and weighted Frobenius bounds are derived. The second main result characterizes exponential stability of the homogeneous Sylvester operator within a separable weighted metric: the natural condition α(A)+α(B)<0 is equivalent to the existence of positive definite weights P,Q for which the weighted logarithmic growth exponent is negative. The weights are constructed from two matrix Lyapunov equations after the shift A↦A+ηI, B↦B−ηI, so an nm×nm Lyapunov equation is avoided even when A or B is not individually Hurwitz. This free shift can also be tuned to tighten finite-time bounds. We further obtain shifted convergence estimates, a weighted nonautonomous extension, and a residual-based a posteriori certificate that contains known Euclidean constant-coefficient residual bounds as a special case. The Lyapunov specialization is treated separately, where Loewner-order bounds again become meaningful. Numerical experiments illustrate the bounds and separable-metric construction.