Matrix Analysis Problems arising in Fast Implementations of the Quantum Monte Carlo Method1

two symmetric matrices, where one matrix has a restricted rank. A comparison of the two cases reveals a remarkable similarity between Eckhart-Young theorem and Ky Fan’s maximum principle. The extremum principles of Ky Fan consider sums and products of eigenvalues of Hermitian matrices. The third part of the talk derives extended rules for rectangular matrices. In this case singular values take the role of eigenvalues. Yet, as we shall see, there are interesting differences between the two cases. The proofs are based on ’rectangular’ versions of Cauchy interlace theorem and Poincaré separation theorem. The minimum norm approach, which motivates the Rectangular quotient, enables us to retrieve a left singular vector from a right one, and vice versa. The ultimate retrieval rule is both a minimum norm solution and a result of an orthogonalization process: Given an estimate, u, of a left singular vector, then the corresponding right singular vector is obtained by orthogonalizing the columns of A against u. Similarly, given an estimate, v, of a right singular vector, then the corresponding left singular vector is obtained by orthogonalizing the rows of A against v. These observations lead to a new orthogonalization method, one that is called “Orthogonalization via Deflation”. The new method constructs an SVD-type decomposition, which is useful for calculating low-rank approximations of large sparse matrices. Another favourable situation arises in problems with missing data. If time allows I will outline the details of the new scheme.