What is a vector Hankel determinant
The aim of the paper under review is to give, under some assumptions, a necessary and sufficient condition that the following system \[ \sum_{j=1}^n x_j a_{i,j} = b_i, \] where \(i=1,\dots,n\), and the \(a_{i,j}\)'s and the \(b_i\)'s are in a euclidean vector space \(V\) of dimension \(n\), has one and only one solution. The author focuses his attention on Hankel system, namely \[ \begin{aligned} x_1 c_0 + x_2c_1+ \cdots +x_nc_{n-1} &= b_1,\\ x_1 c_1 + x_2c_2+ \cdots +x_nc_n &= b_2,\\ & \vdots\\ x_1 c_{n-1} + x_2c_n+ \cdots +x_nc_{2n-2} &= b_n, \end{aligned} \] where \(c_i\), \(i=0,\dots,2n-2\) and \(b_i\), \(i=1,\dots,n\) belong to \(V\). Such system is a vector analogue of the Padé approximation. The author constructs a vector Hankel determinant associated to the previous system \[ \Delta=\left|\begin{matrix} c_0&\cdots&c_{n-1}\\ \vdots& &\vdots\\ c_{n-1}&\cdots&c_{2n-2}\end{matrix}\right|. \] This construction is based on the designant; the right designant [\textit{A. Heyting}, Math. Ann. 98, 465--490 (1927; JFM 53.0120.01)] of a square matrix with elements in a division algebra is the last number to be obtained in a process of reduction to triangular form. The author shows that \(\Delta\) is an element of \(V\) (Theorem 1) and that if \(\Delta\) exists then the Hankel system has one and only one solution \(\bar{x}\in C(V)^n\), \(C(V)\) being the universal real Clifford algebra associated to \(V\), if and only if \(\Delta\) is different from zero (Theorem 2). Most of the proofs involve as a cornerstone some properties of designants that the author has already used in a previous paper [Numer. Algorithms 7, No. 2--4, 225--251 (1994; Zbl 0812.65003)]. The reader is invited to look at the corresponding review of \textit{P. Wynn} (Zbl 0812.65003) on this specific topic.
scientific article; zbMATH DE number 1354776
scientific article