The notion of a transfer (or T -transform) is central in the theory of majorization. For instance, it lies behind the characterization of majorization in terms of doubly stochastic matrices. We introduce a new type of majorization transfer called L-transforms and prove some of its properties. Moreover, we discuss how L-transforms give a new perspective on Ryser’s algorithm for constructing (0; 1)-matrices with given row and column sums.
Let Mm×n(F) be the vector space of all m×n matrices over a field F. In the case where m ≥ n, char(F) ≠ 2 and F has at least five elements, we give a complete characterization of linear maps Φ: Mm×n(F) → Mm×n(F) such that spark(Φ(A)) = spark(A) for any A ∈Mm×n(F).
We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one.
When a system of one-sided max-plus linear equations is inconsistent, its right-hand side vector may be slightly modified to reach a consistent one. It is handled in this note by minimizing the sum of absolute deviations in the right-hand side vector. It turns out that this problem may be reformulated as a mixed integer linear programming problem. Although solving such a problem requires much computational effort, it may propose a solution that just modifies few elements of the right-hand side vector,...
This paper investigates the continuity of projection matrices and illustrates an important application of this property to the derivation of the asymptotic distribution of quadratic forms. We give a new proof and an extension of a result of Stewart (1977).
The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.