  The Lasserre hierarchy is a systematic procedure for constructing a sequence
of increasingly tight relaxations that capture the convex formulations used in
the best available approximation algorithms for a wide variety of optimization
problems. Despite the increasing interest, there are very few techniques for
analyzing Lasserre integrality gaps. Satisfying the positive semi-definite
requirement is one of the major hurdles to constructing Lasserre gap examples.
  We present a novel characterization of the Lasserre hierarchy based on moment
matrices that differ from diagonal ones by matrices of rank one (almost
diagonal form). We provide a modular recipe to obtain positive semi-definite
feasibility conditions by iteratively diagonalizing rank one matrices.
  Using this, we prove strong lower bounds on integrality gaps of Lasserre
hierarchy for two basic capacitated covering problems. For the min-knapsack
problem, we show that the integrality gap remains arbitrarily large even at
level $n-1$ of Lasserre hierarchy. For the min-sum of tardy jobs scheduling
problem, we show that the integrality gap is unbounded at level
$\Omega(\sqrt{n})$ (even when the objective function is integrated as a
constraint). These bounds are interesting on their own, since both problems
admit FPTAS.
