Gershgorin-type spectral inclusions for matrices
In this talk I will discuss recently proposed sequences of Gershgorin-type inclusion sets for the spectra and pseudospectra of finite matrices. In common with previous generalisations of the classical Gershgorin bound for the spectrum, our inclusion sets are based on a block decomposition of the matrix. In contrast to previous generalisations that treat the matrix as a perturbation of a block-diagonal submatrix, our arguments treat the matrix as a perturbation of a block-tridiagonal matrix, which can lead to sharp spectral bounds for particular matrix classes, e.g. for large Toeplitz matrices. Our inclusion sets take the form of unions of pseudospectra of square submatrices. In the Hermitian case these pseudospectra are unions of finite intervals centred on the real eigenvalues of these submatrices, and we show results obtained using open-source software written with ChatGPT. This is joint work with Marko Lindner (Hamburg University of Technology).