Companion matrices: Reducibility, numerical ranges and similarity to contractions
Some unitary-equivalence properties of the companion matrices are studied. It is proved that an \(n\times n\) companion matrix is reducible if and only if its eigenvalues are \(a\omega_n^{j_1},\dots,a\omega_n^{j_p}\), \((1/\overline{a})\omega_n^{j_{p+1}}, \dots,(1/\overline{a})\omega_n^{j_{n}}\), where \(a\neq0\), \(\omega_n\) is the \(n\)th primitive root of \(1\) and \(\{j_1,\dots,j_p\}\), \(\{j_{p+1},\dots,j_n\}\) form a partition of \(\{0,1,\dots,n-1\}\). As a corollary a sufficient condition for a companion matrix to be unitary is obtained. The question in what extent a companion matrix \(A\) is determined by its numerical range \(W(A)\) is also studied. It is proved, e.g., that for \(3\times 3\) companion matrices: if \(W(A)=W(B)\) is not a non-circular elliptic disc, then \(A=B\). The paper is concluded with an improvement of a classical result of \textit{G.-C. Rota} [cf. \textit{P. R. Halmos}, A Hilbert space problem book. Second ed. (1982; Zbl 0496.47001); Corollary 2 to Problem 153] on the similarity of a matrix to a contraction: An \(n\times n\) matrix with spectrum in the open unit disc and multiplicity \(\mu\) is similar to a contraction with defect index \(k\) if and only if \(\mu\leq k\leq n\).
scientific article; zbMATH DE number 2081928
scientific article