Numerical Methods for Solving Inverse Eigenvalue Problems for Nonnegative Matrices
Robert Orsi (RSISE, ANU)
MSI Advanced Computation SeminarDATE: 2006-05-15
TIME: 11:00:00 - 12:00:00
LOCATION: GD 35 (John Dedman building)
CONTACT: JavaScript must be enabled to display this email address.
ABSTRACT:
Inverse eigenvalue problems for matrices come in many forms but they all require one to construct a matrix from prescribed spectral data subject to additional structural constraints on the matrix. In the case of the inverse eigenvalue problem for nonnegative matrices, the additional structural constraints are that the matrix entries be nonnegative, and the matrix many also be required to be stochastic or symmetric.
Finding necessary and sufficient conditions for a list of numbers to be realizable as the eigenvalues of a nonnegative matrix has been a challenging area of research for over fifty years and this problem is still unsolved. While various necessary or sufficient conditions exist, the necessary conditions are usually too general while the sufficient conditions are too specific. Under a few special sufficient conditions, a nonnegative matrix with the desired spectrum can be constructed, however, in general, proofs of sufficient conditions are non-constructive.
This talk will present numerical methods for solving the nonnegative
inverse eigenvalue problems mentioned above. These methods are iterative
in nature and utilize 'alternating projection' ideas. In the case that one
would like to find a symmetric matrix, the algorithm is particularly
simple with the main computational component of each iteration being an
eigenvalue-eigenvector decomposition. The talk will include an overview of
algorithm convergence properties, as well as some numerical results. The
ideas presented in the talk should also be applicable to many other
inverse eigenvalue problems.
