On geometric circulant matrices with geometric sequence (Q6586412)

After an introduction where a review of the literature on \(k\)-circulant matrices is given, the author presents and motivates the definition of a related class of matrices defined by \textit{C. Kızılateş} and \textit{N. Tuglu} [J. Inequal. Appl. 2016, Paper No. 312, 15 p. (2016; Zbl 1349.15060)], namely \textit{geometric circulant matrices}, i.e., matrices completely determined by a nonzero complex number \(k \in \mathbb{C}\backslash \{0\}\) and their first row in the sense that they are of the following form: \N\[\N\text{circ}_n\{_{k^*}(c_0, c_1, c_2, \dots, c_{n-1})\} ~=~ \begin{pmatrix} c_0 & c_1 & c_2 & \dots & c_{n-2} & c_{n-1} \\\Nk c_{n-1} & c_0 & c_1 & \dots & c_{n-3} & c_{n-2} \\\Nk^2 c_{n-2} & kc_{n-1} & c_0 & \dots & c_{n-4} & c_{n-3} \\\N\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\\Nk^{n-2} c_2 & k^{n-3} c_3 & k^{n-4} c_4 & \dots & c_0 & c_1 \\\Nk^{n-1} c_1 & k^{n-2} c_2 & k^{n-3} c_3 & \dots & kc_{n-1} & c_0 \end{pmatrix}.\N\]\NMore specifically, the paper examines a special case of geometric circulant matrices, where the entries in the first row form a geometric progression of common ratio \(q \in \mathbb{R}\backslash \{0\}\) and of scale factor \(g \in \mathbb{C}\backslash \{0\}\). In other words, the author considers matrices of the form \N\[\N\text{circ}_n\{_{k^*}(g, gq, gq^2, \dots, gq^{n-1})\},\N\]\Nand studies their determinant, Frobenius norm, as well as bounds for the spectral norm (i.e., the matrix norm induced by the \(\ell_2\)-norm for vectors).\N\NThe main original results obtained and proved in this article generally come in pairs:\N\N\begin{itemize}\N\item An explicit formula (depending on \(k\), \(n\) and \(q\)) is obtained for \(\text{det}(Q)\), where \N\[\NQ = \text{circ}_n\{_{k^*}(1, q, q^2, \dots, q^{n-1})\}.\N\]\NThis formula is then used to obtain an analogous formula for \(\text{det}(G)\), where \N\[\N G=\text{circ}_n\{_{k^*}(g, gq, gq^2, \dots, gq^{n-1})\}.\N\]\N\item The Moore-Penrose inverse of \N\[\N\text{circ}_n\{_{\left(\frac{1}{q^n}\right)^*}(1, q, q^2, \dots, q^{n-1})\}\N\]\Nis obtained for arbitrary natural number \(n>1\). This serves as a springboard for the computation of the Moore-Penrose of \N\[\N \text{circ}_n\{_{\left(\frac{1}{q^n}\right)^*}(g, gq, gq^2, \dots, gq^{n-1})\}\N\]\Nfor arbitrary natural number \(n>1\).\N\N\item The Moore-Penrose inverse of \N\[\N\text{circ}_n\{_{(-1)^*}(1,-1,1, \dots, -1, 1)\}\N\]\Nis obtained for arbitrary odd natural number \(n>1\). Once again, this serves as a springboard for the computation of the Moore-Penrose of \N\[\N\text{circ}_n\{_{(-1)^*}(g,-g,g, \dots, -g, g)\}\N\] \Nfor arbitrary odd natural number \(n>1\).\N\end{itemize}\N\NLastly, and perhaps most interestingly, explicit formulas for the Frobenius norm of \[\text{circ}_n\{_{k^*}(g, gq, gq^2, \dots, gq^{n-1})\}\] are derived, along with upper and lower bounds for their spectral norm.\N\NIt is worth highlighting that the article contains a number of numerical examples to aid understanding.