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Maximal numerical range


Let $A$ be an $d \times d$ matrix. The maximal numerical range of $A$ is the set: \[ W_0(A) = \{ z : A z, \|A \| \|A\|, = 1, ^d \}. \]

This notion was first introduced in [1]. In [2] it was shown that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. More properties about maximal numerical range for specific class of matrices we can find in [3].


  1. [1]J. Stampfli, “The norm of a derivation,” Pacific journal of mathematics, vol. 33, no. 3, pp. 737–747, 1970, [Online]. Available at:
  2. [2]I. M. Spitkovsky, “A note on the maximal numerical range,” arXiv preprint arXiv:1803.10516, vol. 0, 2018, [Online]. Available at:
  3. [3]A. Hamed and I. Spitkovsky, “On the maximal numerical range of some matrices,” The Electronic Journal of Linear Algebra, vol. 34, pp. 288–303, 2018, [Online]. Available at: