Improving formulas for the eigenvalues of finite block-Toeplitz tridiagonal matrices

Highlights：

• The characteristic polynomial of block-TT matrices with commuting and diagonalizable matrix-entries is fully factored by using a Kronecker-product approach.

• Some basic isospectral properties of the Toeplitz-tridiagonal matrices are generalized to block-TT matrices.

• If certain matrix square-root is well-defined, the study of the eigenvalues of proper block-TT matrices (with commuting matrix-entries) is reduced to that of a related symmetric block-TT one. This result is also extended to non-proper block-TT matrices.

• A block-centrosymmetric decomposition for the block tridiagonal matrices is proposed that allows us to characterize the hierarchical Hermitian block-TT ones.

摘要

•The block tridiagonal matrices are reviewed. In particular, block-Toeplitz tridiagonal (block-TT) matrices (common in the applications) are focused. Their well-known applications to partial differential equations, like Poisson’s equation, are also illustrated.•The characteristic polynomial of block-TT matrices with commuting and diagonalizable matrix-entries is fully factored by using a Kronecker-product approach.•Some basic isospectral properties of the Toeplitz-tridiagonal matrices are generalized to block-TT matrices.•If certain matrix square-root is well-defined, the study of the eigenvalues of proper block-TT matrices (with commuting matrix-entries) is reduced to that of a related symmetric block-TT one. This result is also extended to non-proper block-TT matrices.•A block-centrosymmetric decomposition for the block tridiagonal matrices is proposed that allows us to characterize the hierarchical Hermitian block-TT ones.