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This paper deals with the problem of testing whether a large matrix X has a prescribed structure by looking at the magnitude of a displacement matrix D(X) associated with the structure. We provide parameters on the basis of which one can judge whether the problem is well-conditioned or ill-conditioned. It turns out that even for very general structures it is the minimal eigenvalues of positive definite and banded Toeplitz matrices that are the most important of these parameters. [ABSTRACT FROM AUTHOR]