A definition of numerical range of rectangular matrices

Higher rank numerical ranges of rectangular matrices

“…The vector x 0 is called an eigenvector of A with respect to B corresponding to µ 0 , and the set of all eigenvalues of A with respect to B is denoted by σ(A; B). One may see [6] for more details. It is clear that σ(A; B) = {µ ∈ C : dim ker(A − µB) ≥ 1}.…”

“…2 denotes the spectral matrix norm (i.e., the matrix norm subordinate to the Euclidean vector norm), and I n is the n × n identity matrix. By this idea, Chorianopoulos, Karanasios and Psarrakos [6] recently introduced a definition of the numerical range for rectangular complex matrices. For any A, B ∈ M n×m with B = 0, and any vector norm .…”

“…), which will be useful in our discussion. One may see [6] and [7] for more details. Recall that, in a complex normed space (X, • ), for any ∈ [0, 1), two vectors φ and ψ are said to be Birkhoff-James -orthogonal, denoted by ϕ ⊥ BJ ψ, if ϕ + λψ ≥ √ 1 − 2 ϕ for all λ ∈ C. For the case = 0, we write ϕ ⊥ BJ ψ instead of ϕ ⊥ 0 BJ ψ.…”