Copositive and Positive Quadratic Forms on Matrices

被引：0
作者：
Mohammad Al-khlyleh
Mowaffaq Hajja
机构：
[1] Al-Balqa Applied University,Department of Applied Science, Ajloun College
[2] Philadelphia University,Department of Basic Sciences and Mathematics, Faculty of Science
来源：
Results in Mathematics | 2019年 / 74卷
关键词：
Symmetric matrix; positive semidefinite matrix; positive form; copositive form; test tuples; eigenvalues; 15A45; 15A63; 15B57;
D O I：
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摘要：
A real symmetric quadratic form f=f(Z1,…,Zn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f = f(Z_1,\ldots ,Z_n)$$\end{document} in the n non-commuting indeterminates Z1,…,Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_1,\ldots ,Z_n$$\end{document} is said to be d-positive (respectively, d-copositive) if for all real symmetric (respectively, positive semidefinite) (d×d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d \times d)$$\end{document}-matrices A1,…,An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_1,\ldots ,A_n$$\end{document}, the matrix f(A1,…,An)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(A_1,\ldots ,A_n)$$\end{document} is positive semidefinite. When d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document}, i.e., when Zi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_i$$\end{document} can take real numbers as values, simple characterizations of real positive and copositive symmetric quadratic forms are given, for example, by Hajja (Math Inequal Appl 6:581–593, 2003). In this paper, similar characterizations are obtained for all d.
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