Structured inverse least-squares problem for structured matrices
Provides a complete theoretical solution to a fundamental optimization problem for structured matrices, benefiting researchers in matrix theory and numerical linear algebra.
The paper solves the structured inverse least-squares problem for structured matrices, finding all solutions and those with smallest norm. It shows infinitely many minimal-norm solutions for spectral norm but a unique one for Frobenius norm.
Given a pair of matrices X and B and an appropriate class of structured matrices S, we provide a complete solution of the structured inverse least-squares problem $min_{A\in_S} \|AX-B\|_F$. Indeed, we determine all solutions of the structured inverse least squares problem as well as those solutions which have the smallest norm. We show that there are infinitely many smallest norm solutions of the least squares problem for the spectral norm whereas the smallest norm solution is unique for the Frobenius norm.