Let \(\varOmega_n\) denote the spectral ball, i.e. the set of all \(n\times n\) complex matrices \(A\in\mathcal M_n\) for which \(\rho(A)<1\), where \(\rho(A)\) stands for the spectral radius, \(\rho(A)=\max\{|\lambda|: \lambda\in\text{sp}(A)\}\). The characteristic polynomial \(P_A(t):=\det(tI-A)\) of an \(A\in\mathcal M_n\) may be written in the form \(P_A(t)=t^n+\sum_{j=1}^n(-1)^j\sigma_j(A)t^{n-j}\). Let \(\sigma:=(\sigma_1,\dots,\sigma_N):\mathcal M_n\longrightarrow\mathbb C^n\). Then \(\sigma(\varOmega_n)=\mathbb G_n:=\) the symmetrized polydisk. Let \(\mathcal C_n\) denote the set of all cyclic matrices \(A\in\mathcal M_n\). The main results of the paper are following.
(a) If \(V\in\mathcal C_n\), then \(g_{\varOmega_n}(V,M)=g_{\mathbb G_n}(\sigma(V),\sigma(M))\), \(M\in\mathcal M_n\), where \(g_{\varOmega_n}\) (resp. \(g_{\mathbb G_n}\)) denotes the Green function for \(\varOmega_n\) (resp. \(\mathbb G_n\)). If \(V\notin\mathcal C_n\), then there exists an \(X\in\mathcal M_n\setminus\{0\}\) such that \(g_{\varOmega_n}(V,V+\zeta X)\geq m(\lambda)\log|\zeta|+O(1)\) while \(g_{\mathbb G_n}(\sigma(V), \sigma(V+\zeta X))\leq n(\lambda)\log|\zeta|+O(1)\), where \(m(\lambda)\) (resp. \(n(\lambda)\)) denotes the multiplicity of the eigenvalue \(\lambda\) as a root of the minimal (resp. characteristic) polynomial of \(V\).
(b) For \(A, M\in\varOmega_n\) the following conditions are equivalent: (i) \(g_{\varOmega_n}(A,M)=g_{\mathbb G_n}(\sigma(A), \sigma(M))\), (ii) \(g_{\varOmega_n}\) is continuous at \((A,M)\), (iii) \(g_{\varOmega_n}(\cdot,M)\) is continuous at \(A\).
(c) For \(n\geq3\) we have \(\gamma_{\mathbb G_n}(0;e_{n-1})<A_{\mathbb G_n}(0;e_{n-1})\), where \(\gamma_{\mathbb G_n}\) (resp. \(A_{\mathbb G_n}\)) denotes the Carathéodory–Reiffen (rep. Azukawa) metric for \(\mathbb G_n\).