Numerical ranges of reducible companion matrices
For any complex polynomial \(p(z)= z^n+ a_1 z^{n-1}+\cdots+ a_{n-1} z+ a_n\), its associated companion matrix is \[ \left[\begin{pmatrix} 0 & 1\\ & 0 & 1\\ &&\ddots &\ddots\\ &&& 0 & 1\\ -a_n & -a_{n-1} &\cdots & -a_2 & -a_1\end{pmatrix}\right]. \] The study of the numerical ranges of such matrices was started in two papers by \textit{H.-L. Gau} and \textit{P. L. Wu} [Linear Algebra Appl. 383, 127--142 (2004; Zbl 1063.15025), ibid. 421, No.~2--3, 202--218 (2007; Zbl 1128.15017)]. It is known that there are two 3-by-3 nonequal companion matrices whose numerical ranges are the same elliptic disc. Here the author proves that two \(n\)-by-\(n\) reducible companion matrices with the same numerical range must be equal. (A matrix is reducible if it is unitarily equivalent to the direct sum of two other matrices.) This is done by showing that every such matrix is either unitary or unitarily equivalent to a direct sum \(A\oplus B\), where \(A\) is of class \(S_k\) and \(B\) is of class \(S^{-1}_{n-k}\) for some \(k\), \(1\leq k\leq n-1\). An \(n\)-by-\(n\) matrix \(A\) (resp., \(B\)) is said to be of class \(S_n\) (resp., of class \(S^{-1}_n\)) if all its eigenvalues have moduli less than (resp., larger than) one and rank \((I_n- A^*A)= 1\) (resp., rank \((I_n- B^*B)= 1\)). It is easily seen that an \(S_n\)-matrix must have norm equal to one and an invertible matrix \(B\) is of class \(S^{-1}_n\) if and only if \(B^{-1}\) is of class \(S_n\). As for \(S_n\)-matrices, two \(S^{-1}_n\)-matrices are unitarily equivalent if and only if their numerical ranges are equal. The paper concludes with a characterization of those reducible companion matrices whose numerical ranges are elliptic discs.
scientific article; zbMATH DE number 5669182
scientific article