On a Result of S. Sherman concerning Doubly Stochastic Matrices
نویسنده
چکیده
He also defines, following Hardy, Littlewood and Polya [3] a partial ordering of w-dimensional real vectors: a<b if, and only if, there exists a d.s. matrix P such that a = Pb. In the above named article, the author investigates a conjecture of S. Kakutani to the effect that, if two d.s. matrices are such that Pi& <P3a for every real vector a, then Pi<P3. For this purpose he constructs a linear mapping of all vectors of the form P3a onto vectors Pia, and extends this mapping to a mapping \p of the whole euclidean «-space onto the range of Pi. However, A. Horn (as quoted in [2]) points out, with the aid of a counter-example, that such an extension does not always preserve the properties of a mapping effected by a d.s. matrix, and thus Kakutani's conjecture is not true without restriction. On the other hand, if every vector is of the form P3a, i.e. if the matrix P3 is regular, it is readily verified that Sherman's construction has the required properties and thus in this case, Kakutani's conjecture holds true. It is the object of the present note to give an elementary proof of this fact, as well as to derive a more or less intuitive description of Sherman's partial ordering of d.s. matrices.
منابع مشابه
Some results on the symmetric doubly stochastic inverse eigenvalue problem
The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$, to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$. If there exists an $ntimes n$ symmetric doubly stochastic ...
متن کاملDouble-null operators and the investigation of Birkhoff's theorem on discrete lp spaces
Doubly stochastic matrices play a fundamental role in the theory of majorization. Birkhoff's theorem explains the relation between $ntimes n$ doubly stochastic matrices and permutations. In this paper, we first introduce double-null operators and we will find some important properties of them. Then with the help of double-null operators, we investigate Birkhoff's theorem for descreate $l^p$ sp...
متن کاملEla Some Subpolytopes of the Birkhoff Polytope∗
Some special subsets of the set of uniformly tapered doubly stochastic matrices are considered. It is proved that each such subset is a convex polytope and its extreme points are determined. A minimality result for the whole set of uniformly tapered doubly stochastic matrices is also given. It is well known that if x and y are nonnegative vectors of R and x is weakly majorized by y, there exist...
متن کاملMinimization of Norms and the Spectral Radius of a Sum of Nonnegative Matrices Under Diagonal Equivalence
We generalize in various directions a result of Friedland and Karlin on a lower bound for the spectral radius of a matrix that is positively diagonally equivalent to a • The research of these authors was supported by their joint grant No. 90-00434 from the United States-Israel Binational Science Foundation, Jerusalem, Israel. t The research of this author was supported in part by NSF Grant DMS-...
متن کاملA note on doubly stochastic graph matrices
A sharp lower bound for the smallest entries, among those corresponding to edges, of doubly stochastic matrices of trees is obtained, and the trees that attain this bound are characterized. This result is used to provide a negative answer to Merris’ question in [R. Merris, Doubly stochastic graph matrices II, Linear Multilin. Algebra 45 (1998) 275–285]. © 2005 Elsevier Inc. All rights reserved....
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010