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On the zero set of the Kobayashi--Royden pseudometric of the spectral unit ball

Given $A\inΩ_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a "generalized" tangent vector at $A$ to an entire curve in $Ω_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $Ω_3,$ the zero set of this metric is completely described.