  Uncertainty relations are old, yet potentially rewarding to explore. By
introducing a quantity called the uncertainty matrix, we provide a link between
purity and observable incompatibility, and derive several stronger uncertainty
relations in both forward and reverse directions for arbitrary quantum states,
i.e., mixed as well as pure, and arbitrary incompatible quantum observables,
none of which suffer from the problem of triviality. Besides the tightness, the
interpretations of terms in these uncertainty relations may be of independent
inter- est. We provide the possible generalization of stronger uncertainty
relations to sum of variances of more than two observables. We also demonstrate
applications of techniques used here to, firstly, obtain a simple reverse
quantum speed limit for quantum states undergoing Markovian dynamical
evolution, and secondly, to provide a lower bound for fidelity between two
quantum states.
