Über das Minimum einer gewissen Linearform. (Russian. German summary) Zbl 0083.00501

The author considers real \(m\times n\) matriices (denoted by capitals) and defines
\[ (U,V)=\sum_{i=1}^m \sum_{i=1}^n u_{i,j} v_{ij}, \] \(u\sim V\) if \(\sum_{j=1}^n (u_{i,j} -v_{ij}) = 0= \sum_{i=1}^m (u_{i,j} -v_{ij})\) and \(A\geq 0\) if \(a_{ij}\geq 0\). If \(A\geq 0\) the matrix \(A = (a_{ij})\) is said to be \(K\)-minimal if \((K, A)\leq (K, X)\) for all \(X\sim A\) and \(X\geq 0\). The author defines a matrix \(D\) to be reducible if there exists a matrix \(C\neq D\) where the \(c_{ij}\) are integers and \(0\leq c_{ij}\leq d_{ij}\) or \(0\geq c_{ij}\geq d_{ij}\), otherwise \(D\) is irreducible. If \(F\sim O\), the zero matrix, then \(F\) may be expressed as a sum of positive multiples of irreducible matrices. This result is used to show that \(A\) is \(K\)-minimal if, and only if, \((K,C)\geq 0\) for the set of all integer, irreducible \(C\) for which \(C\sim O\) and \(A+ \rho C\geq 0\) for some \(\rho > 0\). If \(E_{rs}\) is zero everywhere except for unity in column \(s\) of row \(r\), let \(\mathfrak C_A^{(r,s)}\) (with \(A\geq 0)\) denote the set of irreducible, integer matrices \(C\) for which \(C\sim -E_{1r}+E_{1s}\), and \(A+\rho C\geq 0\) for some \(\rho> 0\). If \(A\) is also \(K\)-minimal then \(A+\sigma C\) \((\sigma>0)\) is \(K\)-minimal if, and only if, \((K,C) \leq (K,D)\) for all \(D\) in \(\mathfrak C_A^{(r,s)}\). A method of finding these matrices is given. Finally, if \(B\geq 0\) and the \(b_{ij}\) are rational then a \(U\) can be found such that \(U\sim B\) and \(U\) is \(K\)-minimal.
Reviewer: F. W. Ponting


15A99 Basic linear algebra
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