Generalized matrix nearness problems

09/29/2022

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by   Zihao Li, et al.

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We show that the global minimum solution of ‖ A - BXC ‖ can be found in closed-form with singular value decompositions and generalized singular value decompositions for a variety of constraints on X involving rank, norm, symmetry, two-sided product, and prescribed eigenvalue. This extends the solution of Friedland–Torokhti for the generalized rank-constrained approximation problem to other constraints as well as provides an alternative solution for rank constraint in terms of singular value decompositions. For more complicated constraints on X involving structures such as Toeplitz, Hankel, circulant, nonnegativity, stochasticity, positive semidefiniteness, prescribed eigenvector, etc, we prove that a simple iterative method is linearly and globally convergent to the global minimum solution.