A condensed representation of almost normal matrices

Block tridiagonal reduction of perturbed normal and rank structured matrices

“…Indeed, if A is an almost-normal matrix such that ∆(A) = CA − AC then B = A + γI, with a complex constant γ, is almost normal as well and ∆(B) = ∆(A) = CB − BC. In [1] the structure of almost normal matrices with rank-one perturbation is studied by showing that a blocktridiagonal matrix with 2 × 2 blocks can be determined to satisfy (1) for a certain C of rank one. Here we take a different look at the problem by asking whether a solution of (1) for a given C can be reduced to block tridiagonal form.…”

“…In [1] the class of almost normal matrices is introduced, that is the class of matrices for which [A, A H ] = A H A − AA H = CA − AC for a low rank matrix C. In the framework of operator theory conditions upon the commutator [A, A H ] are widely used in the study of structural properties of hypernormal operators [18]. Our interest in the class of almost normal matrices stems from the analysis of fast eigenvalue algorithms for rank-structured matrices.…”

“…In [1] it is shown that we can always find an almost normal block tridiagonal matrix with blocks of size 2 which fulfills the commutator equation for a certain C with rank(C) = 1. Although an algorithmic construction of a block tridiagonal solution is given, no efficient computational method is described in that paper for the block tridiagonal reduction of a prescribed solution of the equation.…”