[Paper Review] On the convexity and circularity of the numerical range of nilpotent quaternionic matrices

Luís Carvalho, Cristina Diogo+1 more|arXiv (Cornell University)|Jul 31, 2019

Matrix Theory and Algorithms4 citations

TL;DR

This paper establishes sufficient and necessary conditions for the numerical range of nilpotent quaternionic matrices to be circular or convex. It proves that for 3×3 nilpotent matrices, the numerical range is circular if and only if the matrix is cycle-free, and convex if and only if the product $ a_{13}^*a_{12}a_{23} $ is real. The results extend to general nilpotent matrices via graph-theoretic conditions on cycle structure and apply Berge’s maximum theorem to derive convexity for matrices with real diagonal and tree-structured nilpotent parts.

ABSTRACT

We provide a sufficient condition for the numerical range of a nilpotent matrix N to be circular in terms of the existence of cycles in an undirected graph associated with N. We prove that if we add to this matrix N a diagonal real matrix D, the matrix D + N has convex numerical range. For 3 x 3 nilpotent matrices, we strength further our results and obtain necessary and sufficient conditions for circularity and convexity of the numerical range.

Motivation & Objective

• To determine when the numerical range of a nilpotent quaternionic matrix is circular or convex.
• To characterize the shape of the numerical range in the quaternionic setting, where the Toeplitz-Hausdorff theorem does not hold.
• To extend results from the complex case to quaternions, particularly for nilpotent matrices.
• To provide a graph-theoretic condition (cycle-freeness) for circularity and a real product condition for convexity in 3×3 matrices.
• To prove that adding a real diagonal matrix to a nilpotent matrix yields a matrix with convex numerical range.

Proposed method

• Define the numerical range $ W(A) $ as the set $ \{ \mathbf{x}^* A \mathbf{x} \mid \|\mathbf{x}\| = 1 \} $ in $ \mathbb{H}^n $, using quaternionic inner products.
• Use an undirected graph associated with the matrix to detect cycles; a cycle-free graph implies circular numerical range.
• Apply Berge’s maximum theorem to analyze the supremum of the real part of $ \mathbf{x}^* A \mathbf{x} $, enabling convexity proofs.
• Use unitary equivalence to transform matrices into real forms when $ a_{13}^*a_{12}a_{23} \in \mathbb{R} $, leveraging known convexity results for real matrices.
• Employ contradiction arguments via parametrized vectors $ \mathbf{y} \in \mathbb{S}_{\mathbb{H}^3} $ to prove necessity of the real product condition.
• Use the identity $ \pi_{\mathbb{R}}(ab) = \pi_{\mathbb{R}}(ba) $ to analyze reality of expressions involving quaternions.

Experimental results

Research questions

• RQ1 Under what conditions is the numerical range of a nilpotent quaternionic matrix circular?
• RQ2 What is the necessary and sufficient condition for the numerical range of a 3×3 nilpotent quaternionic matrix to be convex?
• RQ3 How does the presence of cycles in the associated graph of a nilpotent matrix affect the shape of its numerical range?
• RQ4 Can the numerical range of a matrix $ D + N $, where $ D $ is real diagonal and $ N $ is nilpotent, be guaranteed to be convex?
• RQ5 What is the relationship between the product $ a_{13}^*a_{12}a_{23} $ and the convexity of the numerical range in 3×3 nilpotent matrices?

Key findings

• For a 3×3 nilpotent matrix $ A $, the numerical range $ W(A) $ is circular if and only if $ A $ is cycle-free.
• The numerical range $ W(A) $ of a 3×3 nilpotent matrix is convex if and only if $ a_{13}^*a_{12}a_{23} \in \mathbb{R} $.
• If $ a_{13}^*a_{12}a_{23} \in \mathbb{R} $, then $ A $ is unitarily equivalent to a real matrix, and hence $ W(A) $ is convex.
• When the nilpotent part of a matrix is tree-structured and the diagonal part is real, the numerical range is a union of disks, and thus convex.
• The paper provides an example where the union of disks in the numerical range forms an ellipse, showing that convexity does not imply circularity.
• For general nilpotent matrices, a sufficient condition for circularity is that the associated graph is a tree, though this condition is not necessary.