Elimination techniques and interpolation
Let K be an arbitrary field and let V be a vector space over K. Let \(A=(a_{ij})_{1\leq i,j\leq n+m}\in K^{n+m,n+m}\). Assume that there exist \((1\leq s\leq m+1)\)-regular square submatrices \(A_ h\) (h-1,2,...,s) in the n first columns of A. We denote by \(a_{1h}<a_{2h}<...<a_{nh}\) (h-1,2,...,s) the indices of the rows of A and write \(I_ h=\{a_{1h},a_{2h},...,a_{nh}\}\subset \{1,2,...,n+m\}\), and \(J=\{1,2,...,n+m\}-\cup^{s}_{h=1}I_ h\). Let \(\{v_ 1,v_ 2,...,v_{n+m}\}\) be a set of linearly independent elements of V and \(L_ i\) be a set of \(n+m\) linear forms on V such that \(L_ iV_ j=a_{ij}\), \(0\leq i\leq n+m\), \(1\leq j\leq n+m\). Let us now suppose that \((1)\quad Card\cup^{s}_{h=1}I_ h=n+s-1,\quad Card(I_ n\cap I_{h+1})=n-1,\) \(1\leq h\leq s-1\), \(\det (a_{it})_{1\leq t\leq n,i\in \{0\}\cup (I_ h\cap I_{h+1})}\neq 0\), and \[ D = \left[ \begin{matrix} 1 & 1 & \ldots & 1 \\ a_{01}^1 & a_{01}^2 && a_{01}^ s \\ \vdots & \vdots & && \vdots \\ a_{0(s-1)}^1 & a_{0(s-1)}^2 && a_{0(s-1)}^ s \end{matrix} \right] \left[ \begin{matrix} 1 & 1 & \ldots & 1 \\ L_0 \gamma_{n,1}^{n+1} & L_0 \gamma_{n,2}^{n+1} & \ldots & L_0 \gamma_{n,s}^{n+1} \\ \vdots & \vdots & && \vdots \\ L_0 \gamma_{n,1}^{n+s-1} & L_0 \gamma_{n,2}^{n+s-1} & \ldots & L_0 \gamma_{n,s}^{n+s-1} \\ \end{matrix} \right] \] In the main result of the paper the authors show that if \(D\neq 0\), \(s<m+1\) and the conditions in (1) hold, then A is regular iff \(\det (L_ t\gamma^{n+j}_{n+s-1})_{t\in J,s\leq j\leq m}\neq 0,\) where \(\gamma^{n+j}_{n+s-1}=v_{n+j}-p^{n+j}_{n+s-1},\) \(s\leq j\leq m\), \(p^{n+j}_{n+s-1}\in V_{n+s-1},\quad L_ tp^{n+j}_{n+s-1}=L_ tv_{n+j},\) \(t\in \cup^{s}_{h=1}I_ h\) and \(V_ n=span\{v_ 1,v_ 2,...,v_ n\}\).
scientific article; zbMATH DE number 4065578
scientific article