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A direct proof of the complete Crouzeix inequality

OpenAI

OpenAI Mathopenai-math/A-direct-proof-of-the-complete-Crouzeix-inequalitySep 26, 2026math.FAAI-generatedscore 0

We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.

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