在全非負矩陣上的哈達馬運算

Hadamard Operations on Totally Nonnegative Matrices

林志嘉

指導教授 ： 翁志文

國立交通大學/理學院/應用數學系暨研究所/碩士(2011年)

https://doi.org/10.6842/NCTU.2011.00121

摘要

本論文我們關心的是全非負矩陣和全正矩陣的性質。一個其任意子矩
陣的行列式值皆為非負或皆為正的非負矩陣稱之為全非負矩陣或為全正矩陣。這樣的矩陣在數學領域裡扮演著相當重要的角色。此篇論文主要的目的在於介紹非負矩陣和正矩陣的基本性質及收集一些關於全非負矩陣和全正矩陣的已知性質，並重新回憶哈達馬乘積和半正定性質的關聯。此外，我們也將討論關於永恆全正的性質。

關鍵字

全非負矩陣

並列摘要

In this thesis we are concerned with the properties of totally nonnegative ( resp. positive ) matrices. An m-by-n entry-wise nonnegative matrix is totally nonnegative (resp. positive) if the determinant of any square submatrix is
nonnegative (resp. positive). A totally nonnegative matrix plays an important role in various mathematical branches. The primary propose here is to introduce the basic properties of nonnegative (resp. positive) matrix and to collect the known results in matrix theory with the totally nonnegative (resp. positive) property involved. We will recall the relation between Hadamard product and the positive semidefinite property, and study the relation between Hadamard product and the totally nonnegative (resp. totally positive) property. Furthermore, we also discuss the eventually totally positive property.

並列關鍵字

toatlly nonnegative matrices

參考文獻

[1] T. Ando, Totally positive matrices, Linear Algebra Appl. 90(1987)165–219.

[4] A.S. Crans, S.M. Fallat, C.R. Johnson, The Hadamard core of the totally nonnegative matrices, Linear Algebra Appl. 328(2001) 203–222.

[5] Shaun M. Fallat, Charles R. Johnson, Hadamard powers and totally positive matrices, Linear Algebra and its Applications. 423(2007) 420–427.

[7] Minc, Henryk, Nonnegative matrices, New York, Wiley, 1988.

[8] Vern I. Paulse, Stephen C. Power, Roger R. Smith, Schur Products and Matrix Completions, Journal of Functional Analysis. 85 (1989)151–178.