The talk will consist of two parts. The first one based on the joint work with S.Fallat and C.R.Johnson, deals with a problem of describing all possible determinantal inequalities for principal minors of totally nonnegative matrices. These inequalities generalize classical inequalities of Hadamard, Fischer, Szasz etc. I will describe all such inequalities of classical type, explain methods involved in complete solution of the problem for low dimensions and formulate a number of conjectures.

In the second part (based on joint work with L.Faybusovich) results on non-standard inverse problems for certain classes of Hessenberg and symmetric matrices will be presented. These inverse problems can be viewed as a generalization of both Hankel and Toeplitz moment problems and can be applied in a study of completely integrable lattices obtained as restrictions of full Toda flows to low-dimensional symplectic leaves.

Existence of a factorization into elementary bi-diagonal factors plays an important role in both parts of the talk.