Extended Numerical Range of Matrix Entries

• Extended numerical range of matrix entries is a generalization of the classical numerical range that incorporates higher-rank compressions, polynomial matrices, and nonstandard inner products.
• It leverages convex geometric and algebraic characterizations, revealing phenomena such as dimensional collapse, boundary sharp points, and complex spectral behaviors.
• These insights offer practical benefits in control systems, quantum error correction, and spectral estimation through refined matrix analysis techniques.

The extended numerical range of matrix entries refers to a spectrum of advanced generalizations of the classical numerical range, expanding both the algebraic and geometric frameworks underpinning matrix analysis. Moving far beyond scalar compressions and classical fields, recent research systematically enlarges this notion along multiple axes: higher-rank compressions, structured block and polynomial matrices, nonstandard inner products, algebraic extensions, and even non-classical semirings. This has led to a dramatic enrichment of the theoretical and practical landscape surrounding matrix values, revealed through invariants and geometric sets with deep applications—in operator theory, control, quantum error correction, spectral estimation, and algebraic geometry.

1. Higher-Rank Numerical Ranges and Their Structure

The first major extension generalizes the classical numerical range $F(A) = \{x^* A x : x \in \mathbb{C}^n, \|x\|=1\}$ to higher-rank compressions. The higher-rank numerical range $\Lambda_k(A)$ is defined as: $\Lambda_k(A) = \{ \lambda \in \mathbb{C} \mid \text{there exists a rank-%%%%2%%%% orthogonal projection %%%%3%%%% with %%%%4%%%%} \}$ or equivalently

$\Lambda_k(A) = \{ \lambda \in \mathbb{C} : X^* A X = \lambda I_k\ \text{for some %%%%5%%%% with %%%%6%%%%} \}$

For $k=1$, one recovers the classical field of values, and for $k>1$, this defines a nested family $F(A) \supseteq \Lambda_2(A) \supseteq \cdots$, providing increasingly refined geometric probes of the operator $A$ (Aretaki et al., 2011, Aretaki et al., 2011, Nino-Cortes et al., 29 Oct 2024).

Key properties and implications:

• Convexity and Compactness: Each $\Lambda_k(A)$ is convex and compact (Nino-Cortes et al., 29 Oct 2024).
• Dimensional Collapse: For $k$ large (e.g., $k \geq (n+1)/2$), $\Lambda_k(A)$ may be a line segment, singleton, or even empty, depending on algebraic multiplicities in the spectrum of $A$ (Nino-Cortes et al., 29 Oct 2024).
• Boundary via Kippenhahn Curves: The boundaries of $\Lambda_k(A)$ can be characterized as the envelope of lines determined by the $k$-th eigenvalue of $\operatorname{Re}(e^{i\theta} A)$ as $\theta$ varies, so-called Kippenhahn curves. The convex hull of the union of these curves yields the numerical range, and specific components delineate higher-rank ranges (Bebiano et al., 2021, Nino-Cortes et al., 29 Oct 2024).

2. Extension to Matrix Polynomials and Companion Matrices

The extended numerical range framework naturally generalizes to matrix polynomials: $$L(\lambda) = A_m \lambda^m + \cdots + A_1 \lambda + A_0$$ The higher-rank numerical range for the polynomial, $\Lambda_k(L(\lambda))$, consists of all complex $\lambda$ for which $Q^* L(\lambda) Q = 0_k$ for some isometric $Q$.

Notable results:

• Companion Matrix Transfer: For Perron polynomials (monic with entrywise nonnegative coefficients), one constructs a companion matrix $C_L$ whose higher-rank numerical range encodes that of $L(\lambda)$. Spectral and symmetry properties (including cyclicity and equidistributed maximal elements) transfer from $C_L$ to $L(\lambda)$ (Aretaki et al., 2011).
• Sharp Points: Maximal elements of the extended numerical range coincide with “sharp points,” boundary locations that realize unique supporting directions and are related to eigenvalues of high multiplicity (Aretaki et al., 2011).
• Joint Numerical Range Inclusion: The extended scalar range relates closely to the joint higher-rank numerical range for $(A_0, \ldots, A_m)$, with inclusions established between the scalar and vector-valued versions (Aretaki et al., 2011).

3. Geometric and Algebraic Characterizations

A central advance is the detailed real algebraic and convex geometric analysis of extended numerical ranges:

• Algebraic Boundary: For any $A$, the classical Kippenhahn polynomial $$f_A(t,x,y) = \det(tI + x \operatorname{Re}A + y \operatorname{Im}A)$$ encodes the numerical range as the convex hull of a real plane algebraic curve. For higher-rank and extended constructions, divisibility conditions (e.g., linear factors of high multiplicity) and dual curves (supporting line loci) determine lower-dimensional or degenerate cases (Nino-Cortes et al., 29 Oct 2024, Bebiano et al., 2021).
• p-Tangent and Tri-Tangent Lines: In lower-dimensional cases (singleton, segment), extended numerical ranges correspond to faces of the convex hull annihilated by certain tangency conditions at singular points of the Kippenhahn curve (Nino-Cortes et al., 29 Oct 2024).
• Symmetries and Cyclic Structures: For nonnegative, irreducible, or cyclic matrices (including permutation and shift matrices), rotational symmetry leads to equidistribution of maximal elements around a circle in the complex plane (Aretaki et al., 2011).

4. Connection with Block Structures, Polynomials, and Indefinite Inner Products

Other forms of extension arise in:

• Block Partitioning: The numerical range of a block matrix, or the off-diagonal block (e.g., in positive matrices $$M = \begin{bmatrix} A & X \ X^* & B \end{bmatrix}$$), critically depends on geometric parameters such as the width or distance from zero of $W(X)$, yielding sharp norm and spectral inequalities (Bourin et al., 2020).
• Indefinite Krein Spaces: The Krein space numerical range $W^J(A)$, defined using an indefinite inner product with a signature matrix $J$, may yield non-convex, even hyperbolic structures. Nondegeneracy and explicit shape conditions (e.g., for tridiagonal and centrosymmetric matrices) have been fully characterized in terms of matrix entries (Bebiano et al., 3 Jul 2024).
• Field Extensions and Max Algebra: By replacing the base field with a degree-2 Galois extension, or even considering the max algebra (where addition is replaced by maximization), novel types of extended (e.g., interval-like, or highly algebraically structured) numerical ranges appear; these exhibit, among other phenomena, cases where eigenvalues may be absent from the extended numerical range (Ballico, 2016, Aboutalebi et al., 20 Nov 2024).

5. Algorithmic, Spectral, and Quantum Applications

Beyond their intrinsic geometric and algebraic importance, extended numerical ranges underpin several computational and applied advances:

• Algorithmic Membership and Boundary Computation: For general $A$, precise algorithms are available to check membership in $\Lambda_k(A)$, leveraging discriminant, elimination, and cylindrical algebraic decomposition methods (Nino-Cortes et al., 29 Oct 2024).
• Spectral Set Constants: Extended numerical ranges provide sharp spectral norm bounds for polynomial or rational functional calculus (notably, Crouzeix’s constant for spectral sets), with conjectural universal constants well supported numerically (Crouzeix, 2016).
• Stabilization and Control: In control and stochastic processes, extended numerical ranges inform perturbation design (e.g., “hollowizing” diagonal entries via orthogonal transformations) to facilitate stabilization via rotation or noise (Damm et al., 2019).
• Quantum Error Correction: The geometry of higher-rank numerical ranges is directly connected with the existence of quantum error-correcting codes; the non-emptiness of the joint $k$-numerical range for certain operator tuples is a necessary condition for code construction (Aretaki et al., 2011, Nino-Cortes et al., 29 Oct 2024).

6. Recent Generalizations and Special Cases

Emerging themes include:

• Random Matrices: For large random non-normal matrices (Ginibre, triangular, etc.), the numerical range almost surely converges (in Hausdorff metric) to a disk whose radius quantifies non-normality, distinct from the limiting spectrum (Collins et al., 2013).
• Correlation Numerical Range: Defined via averaging over all correlation matrices, this extended range is convex, “modded out” by diagonal perturbations, and connected to questions in operator algebra such as Connes’ embedding problem (Hadwin et al., 2011).
• Cyclic Shift Matrices: The numerical range’s dependence on permutation or perturbation of weights is rigorously characterized, with extremal arrangements yielding largest and smallest extended ranges; conjectures remain on maximization patterns in higher $n$ (Chien et al., 2023).
• Normalized Numerical Range: The normalized numerical range $F_N(A)$, formed via $(x^* A x)/(\|Ax\|\|x\|)$, is generally nonconvex but is obtained as a nonlinear image of the Davis-Wielandt shell; its geometry is explicitly described for $2 \times 2$ and normal matrices and reveals further types of extension (Lins et al., 2017).

7. Outlook and Further Directions

The paper of extended numerical ranges is now deeply interwoven with convex algebraic geometry, operator theory, random matrix theory, quantum information, and the theory of non-commutative or tropical semirings. Central open questions include precise characterizations of lower-dimensional extended numerical ranges, universality of various spectral set constants, and the development of comprehensive computational frameworks for these sets in high dimensions and in structured matrix classes.

The continued interplay between algebraic multiplicity, geometric configuration, and field or semiring structure ensures that the extended numerical range will remain a focal point for advances in both matrix theory and its applications to quantum, stochastic, and control systems.