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On numerical range of normal matrices of quaternions. (English) Zbl 0855.15006

The well known Toeplitz-Hausdorff theorem states that the numerical range of a complex matrix \(A\), defined as \[ W_c (A) = \{x^* Ax : x \in \mathbb{C}^n,\;x^* x = 1\}, \] is a convex subset of \(\mathbb{C}\). If \(\mathbb{C}\) is replaced by the field \(\mathbb{H}\) of the quaternions, that statement is no longer true for the corresponding numerical range, \(W_q (A)\). Moreover, if the complex matrix \(A\) is normal, \(W_c (A)\) is the convex hull of the eigenvalues of \(A\).
In the present paper the author looks for properties of the upper build of \(A\), denoted by \(B_+ (A)\) and defined as the intersection of \(W_q (A)\) with the closed upper-half complex plane. He conjectures that \(B_+ (A)\) is a convex subset of \(\mathbb{C}\), for \(A \in \mathbb{H}^{n \times n}\). This was known to be true for normal quaternion matrices, but the author presents an alternative proof of this result, using the Lagrange multipliers rule and another interesting result, which guarantees that, if \(B_+ (A)\) has the convexity property for \(A \in \mathbb{H}^{2 \times 2}\), then \(B_+ (A)\) is a convex set for \(A \in \mathbb{H}^{n \times n}\), \(\forall n > 2\).
The paper also includes a few open questions concerning quaternion matrices.

MSC:

15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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