On the spectral properties of real antitridiagonal Hankel matrices (Q2105346)

The author gives functional characterizations for the eigenvalues and the eigenvectors of an antitridiagonal matrix. Given a vector \(\mathfrak{a}=(a_1,\ldots,a_n)\in \mathbb{R}^n\) (or \(\mathbb{C}^n\)), the \(n\times n\) anticirculant matrix \(C_{\mathfrak{a}}=[c_{i,j}]\) is defined by \(c_{i,j}=a_{[i+j-1]}\) for \(1\le i,j\le n\) where \([i+j-1]\) is the class of \(i+j-1\) \(\pmod{n}\), for \[C_{\mathfrak{a}}=\begin{pmatrix}a_1&a_2&\ldots&a_n\\ a_2&a_3&\ldots&a_1\\ \vdots &\vdots&\ddots&\vdots\\ a_n&a_1&\ldots&a_{n-1}\end{pmatrix}.\] Set \(\mathfrak{a}_0\) the vector \(\mathfrak{a}\) for which \(a_1=b\), \(a_{n-1}=a, a_n=c\) and \(a_i=0\) otherwise. Let \(R=[r_{i,j}]\) be the rank two \(n\times n\) matrix \[\begin{cases}r_{11}=b, r_{n,n}=a\\ r_{i,j}=0 \quad \text{otherwise.}\end{cases}\] An antitridiagonal matrix \(H\) of dimension \(n\) is written as \(C_{{\mathfrak{a}}_0}-R\). The author applies the eigendecomposition of \(C_{{\mathfrak{a}}_0}\) explicitly to \(H\). The characterization (secular) formula for the eigenvalues is then obtained using results in [\textit{J. Anderson}, Linear Algebra Appl. 246, 49--70 (1996; Zbl 0861.15006)] while the corresponding eigenvector characterization comes from [\textit{J. R. Bunch} et al., Numer. Math. 31, 31--48 (1978; Zbl 0369.65007)].