Positive Classes of Matrices

In this lecture we present different types of positivity concept in matrix analysis.Any kind of positive matrix has own typical applications. Entrywise positivity, definitepositivity, complete positivity and total positivity are the main types of positivity inmatrix analysis. The concept of a positive definite matrix (PD) is well-known for mostpeople having the elementary course in linear algebra, but the other types of positivityare not quiet well-know as PD matrices, thus we present other types of positivitiesin matrix theory. For more complete information on PD matrices see [۱]. In linearalgebra any real matrix with nonnegative entries is called Nonnegative Matrix (NM).A matrix which is both nonnegative and positive semi-definite is called doublynonnegative matrix (DNM). The Perron–Frobenius theorem, proved by Oskar Perron(۱۹۰۷) and Georg Frobenius (۱۹۱۲), is the most important result stating that a realsquare matrix with positive entries has a unique largest real eigenvalue and that thecorresponding eigenvector can be chosen to have strictly positive entries. This theoremhas signifant applications [۲, ۴–۶]. If a symmetric matrix A can be factorized ofthe form A = BBT where B is a non-negative matrix, then A is called a CompletelyMatrix (CP). Completely positive matrices have arisen in some situations in economicmodelling and appear to have some applications in statistics, and they are also appearin quadratic optimisation, for more details see [۳]. Any real matrix with nonnegativeminors are called Totally Non-Negative (TN) matrix. If all minors are strictly positivethen A is called Totally Positive (TP). This topic appears in the spectral propertiesof kernels of ordinary differential equations whose Green’s function is totally positive(studied by M. G. Krein and some colleagues in the mid-۱۹۳۰s) [۷–۹]. In thispresentation we give a detailed picture of all kinds of positivity mentioned above.

کلیدواژه ها:

Positivity ، Total Positivity ، Complete Positivity