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Numerical range of $A$ with respect to $B$


Let $A$ and $B$ be $N \times M$ matrices and let $\| \cdot \|$ be any matrix norm. We define the numerical range of $A$ with respect to $B$, as the compact and convex set [1] \[ W_{\| \cdot \|}(A; B) = \left\{ \mu \in \mathbb{C}: \| A - \lambda B \| \geq |\mu - \lambda|, \forall \lambda \in \mathbb{C} \right\} = \bigcap_{\lambda \in \mathbb{C}} \mathcal{D}(\lambda,\| A - \lambda B \|), \] where $\mathcal{D}(a,r)$ denotes a closed disc on the complex plane with center $a$ and radius $r$.

For a square matrix $C \in \mathbb{1}^{N \times N}$, we get \[ W_{\|\cdot\|_2}(A; \1_N) = W(A), \] where $W(A)$ denotes the standard numerical range.

Numerical range of a square matrix


Let $A$ be an $N \times M$ matrix, with $N > M$ given by $A=\begin{pmatrix}A_1 \\ A_2 \end{pmatrix}$ and let $\1_{N,M} = \begin{pmatrix}\1_M \\ 0\end{pmatrix}$, where $\1_M$ denotes an $M \times M$ identity matrix. The numerical range of $A$ is given by \[ W_{\|\cdot\|_2}(A; \mathbb{1}_{N,M}) = W(A_1), \] where $W(A_1)$ denotes the standard numerical shadow.

1. Ch. Chorianopoulos, S. Karanasios, P. Psarrakos, 2009. A definition of numerical range of rectangular matrices. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.459–475.
numerical-range/generalizations/numerical-range-of-a-with-respect-to-b.txt · Last modified: 2018/10/08 09:21 by plewandowska