## 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

### MSC:

 15A99 Basic linear algebra

### Keywords:

linear forms; minimum
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