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The inverse eigenvalue problem for real symmetric Toeplitz matrices. (English) Zbl 0813.15006
This paper solves the previously open inverse eigenvalue problem for real symmetric Toeplitz matrices: the spectrum of such matrices can be arbitrary real. The proof uses the known fact that all corresponding eigenvectors are either even or odd and their parity properties. The result is actually proven to be true for all regular Toeplitz matrices, i.e., those all of whose principal submatrices have the same odd/even eigenvector and parity properties.
The actual proof is topological and nonconstructive: the degree of the map: \(T_ r \to \sigma (T)\) does not vanish for all regular Toeplitz matrices, which implies the result.
Reviewer: F.Uhlig (Auburn)

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
15A21 Canonical forms, reductions, classification
55M25 Degree, winding number
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