It is known that every complex square matrix with nonnega-tive determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of kpositive semi-definite matrices but not fewer, for k=1, 2, 3, 4, 5.

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Let$G$be a simple and simply-connected complex algebraic group,$P$⊂$G$a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology$QH$^{*}($G$/$P$) of a flag variety is, up to localization, a quotient of the homology$H$_{*}(Gr_{$G$}) of the affine Grassmannian Gr_{$G$}of$G$. As a consequence, all three-point genus-zero Gromov–Witten invariants of$G$/$P$are identified with homology Schubert structure constants of$H$_{*}(Gr_{$G$}), establishing the equivalence of the quantum and homology affine Schubert calculi.

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