On the shell and the shell-extremal eigenvalues of a square matrix

Shell extremal eigenvalues of tridiagonal Toeplitz Matrices

“…Proof. By Corollary 3.14 of [6] we have that n 1,Tn(c,b,a) (λ k ) ≤ 4n 3,Tn(c,b,a) (λ k ). Moreover,…”

“…Measures of non-normality. Motivated by property SE6 in the introduction, the following quantities can be considered as measures of non-normality of a shell extremal eigenvalue λ 0 of A ∈ M n (C) as they vanish if and only if λ 0 is a normal eigenvalue [6].…”

“…An eigenvalue with equal algebraic and geometric multiplicities that its corresponding eigenspace is orthogonal to the eigenspaces of all other eigenvalues is called normal. In [6], focus was given to the case where D(A) > 0 and several geometric aspects of the closed branch of the shell were studied and proposed as measures of non-normality of the extremal eigenvalues of a square matrix A that can be surrounded by closed branches of the shells Γ(e iθ A), θ ∈ [0, 2π]. Definition 1.1.…”

“…Here, we list a series of properties of shell extremal eigenvalues and the closed branches of the shells that surround them [6].…”