Krein spaces numerical ranges and their computer generation

Natalia Bebiano, Joao da Providencia, Ana Cristina Nata, Maria da Graca Pereira Soares
2008 The Electronic Journal of Linear Algebra  
Let J be an involutive Hermitian matrix with signature (t, n − t), 0 ≤ t ≤ n, that is, with t positive and n − t negative eigenvalues. The Krein space numerical range of a complex matrix A of size n is the collection of complex numbers of the form ξ * JAξ ξ * Jξ , with ξ ∈ C n and ξ * Jξ = 0. In this note, a class of tridiagonal matrices with hyperbolical numerical range is investigated. A Matlab program is developed to generate Krein spaces numerical ranges in the finite dimensional case.
more » ... ensional case. Introduction. Throughout, M n denotes the algebra of n × n matrices over the field of complex numbers. Let J be an involutive Hermitian matrix with signature (t, n−t), 0 ≤ t ≤ n, that is, with t positive and n−t negative eigenvalues. Consider C n as a Krein space with respect to the indefinite inner product [ξ, η] = η * Jξ, ξ, η ∈ C n . The J−numerical range of A ∈ M n is denoted and defined by: Considering J the identity matrix of order n, I n , this concept reduces to the well known classical numerical range, usually denoted by W (A). The numerical range of an operator defined on an indefinite inner product space is currently being studied (see [11] and references therein). For W J (A), A ∈ M n , the following inclusion holds: σ(A) ⊂ W J (A), where σ(A) denotes the set of the eigenvalues of A with J−anisotropic eigenvectors, that is, eigenvectors with nonvanishing J−norm. We denote by σ ± (A) the sets of the eigenvalues of A with associated eigenvectors ξ such that ξ * Jξ = ±1. Compactness and convexity are basic properties of the classical numerical range. In *
doi:10.13001/1081-3810.1258 fatcat:yvpq4ghe6ndjfkdmgxdobamvaq