Geodesic
A Simple proof of the Goldberg–Straus theorem on numerical radii
Glasgow mathematical journal, Tome 28 (1986) no. 2, pp. 139-141

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Let Mn(C) be the algebra of n × n complex matrices, and let be its unitary group. Given A, B ε Mn(C), the A-numerical radius of B is the nonnegative quantityIn particular, for A = diag(1, 0, ..., 0) it reduces to the classical numerical radius r(B) = max||x*Bx|:x*x = 1}. In [1] Goldberg and Straus proved that rA is a generalized matrix norm (i.e. a positive definite seminorm) on Mn(C) if and only if A is nonscalar and tr A ≠ 0. This result agrees with the well-known fact that the classical numerical radius r is a generalized matrix norm. The nontrivial part of the proof is to show that if A is nonscalar and tr A ≠ 0 then rA is positive definite; that is, for any B ε Mn(C), tr(AU*BU) = 0 for all U ε implies B = 0. The proof given in [1] is computational and involves the use of differentiation on matrices. Later Marcus and Sandy [2] gave three elementary proofs of the result. Their proofs are still computational in nature and two of them need knowledge of multilinear algebra. 
Tam, Bit-Shun. A Simple proof of the Goldberg–Straus theorem on numerical radii. Glasgow mathematical journal, Tome 28 (1986) no. 2, pp. 139-141. doi: 10.1017/S0017089500006455
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[1] 1.Goldberg, M. and Straus, E. G., Norm properties of C-numerical radii, Linear Algebra and Appl. 24 (1979), 113–131. Google Scholar | DOI

[2] 2.Marcus, M. and Sandy, M., Three elementary proofs of the Goldberg–Straus theorem on numerical radii, Linear and Multilinear Algebra 11 (1982), 243–252. Google Scholar | DOI

[3] 3.Tam, B. S., The action of unitary transforms of a matrix on linear subspaces, submitted for publication. Google Scholar

[4] 4.Tam, T. Y., On the generalized radial matrices and a conjecture of Marcus and Sandy, Linear and Multilinear Algebra, to appear. Google Scholar

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