A note on the spectral distribution of symmetrized Toeplitz sequences
Provides analytical results for the spectral properties of symmetrized Toeplitz matrices, extending known distribution theorems to a new class of structured matrices.
The paper derives the singular value and spectral distribution of symmetrized Toeplitz sequences, showing that under a sparsely vanishing symbol, roughly half of the eigenvalues are negative/positive for large dimensions, indicating asymptotic indefiniteness.
The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szeg{ő} theorem and the Avram-Parter theorem, in which the singular value symbol coincides with the generating function. More general versions of the theorem were later proved by Zamarashkin and Tyrtyshnikov, and Tilli. Considering (real) nonsymmetric Toeplitz matrix sequences, we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically, by using the notion of approximating class of sequences. In particular, under the assumption that the symbol is sparsely vanishing, we show that roughly half of the eigenvalues of the symmetrized Toeplitz matrix (i.e. a Hankel matrix) are negative/positive for sufficiently large dimension, i.e. the matrix sequence is symmetric (asymptotically) indefinite.