Between Geršgorin and minimal Geršgorin sets

The eigenvalues of a given matrix A can be localized by the well-known Geršgorin theorem: they belong to the Geršgorin set, which is the union of the Geršgorin disks (each of them is a simple function of the matrix entries). By applying the same theorem to a similar matrix X-1AX, a new inclusion set can be obtained. Taking the intersection over X, being a (positive) diagonal matrix, will lead us to the minimal Geršgorin set, defined by Varga [R.S. Varga, Geršgorin and His Circles, Springer Series in Computational Mathematics, vol. 36, 2004], but this set is not easy to calculate. In this paper we will take the intersection over some special structured matrices X and show that this intersection can be expressed by the same formula as the eigenvalue inclusion set CS(A) in [L.J. Cvetković, V. Kostić, R. Varga, A new Geršgorin-type eigenvalue inclusion set, ETNA 18 (2004) 73-80].