Total positive matrix

This article deals with totally positive matrices, not to be confused with positive definite matrices and positive matrices .

In mathematics , totally positive matrices are (real) matrices whose minors are all positive. Total positive matrices play a role in various areas of mathematics, such as graph theory , algebraic geometry , stochastic processes , game theory , matroid theory and differential equations , as well as Brownian motion , electrical networks and chemistry .

definition

A - matrix ${\displaystyle n\times n}$

${\displaystyle \mathbf {A} =(A_{ij})_{1\leq i,j\leq n}}$

is called totally positive if all of their minors

${\displaystyle M_{I,J}:=\det \left[\mathbf {A} \right]_{I,J},\ I,J\subset \left\{1,\ldots ,n\right\},\ |I|=|J|}$

(i.e. the determinants of the square sub-matrices that result from deleting one or more columns and rows) are positive:

${\displaystyle M_{I,J}>0\quad {\text{für alle}}\quad I,J\subset \left\{1,\ldots ,n\right\},\ |I|=|J|}$.

Examples

Vandermonde matrices with are totally positive. ${\displaystyle V(x_{1},x_{2},\ldots ,x_{n})}$ ${\displaystyle 0<x_{1}<x_{2}<\ldots <x_{n}}$

Theorem by Gantmacher and Krein

The theorem of Gantmacher and Kerin says that every totally positive matrix becomes a diagonal matrix

${\displaystyle diag(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})}$

with is similar . ${\displaystyle \lambda _{1}>\lambda _{2}>\ldots >\lambda _{n}>0}$

history

Total positive matrices are a special case of positive matrices (matrices with positive entries) that were first examined by Oskar Perron . He proved that the absolute greatest eigenvalue of a positive matrix is ​​real, positive and a simple eigenvalue . Gantmacher and Krein examined totally positive matrices and proved that all of their eigenvalues ​​are real, positive and simple. Anne Whitney proved a reduction theorem with which one can represent the totally positive matrices as a submonoid of generated by an explicit set of simple matrices . This approach was then carried over by Lusztig to any semi-simple Lie groups. ${\displaystyle GL(n,\mathbb {R} )}$

literature

• T. Ando: Totally positive matrices. In: Linear Algebra Appl. Volume 90, 1987, pp. 165-219.
• F. Brenti: The applications of total positivity to combinatorics, and conversely. In: FR Gantmacher, MG Krein: Oscillation matrices and kernels and small vibrations of mechanical systems. AMS Chelsea Publ., Providence, RI 2002, pp. 451-473.
• George Lusztig: Introduction to total positivity. In: Positivity in Lie theory: open problems. (= de Gruyter expositions in mathematics. Volume 26). de Gruyter, Berlin 1998, ISBN 3-11-016112-5 , pp. 133-145.
• S. Fomin, A. Zelevinsky: Total positivity: tests and parametrizations. In: Math. Intelligencer. Volume 22, 2000, pp. 23-33.
• G. Lusztig: A survey of total positivity. In: Milan J. Math. Volume 76, 2008, pp. 125-134.

Web links

• Minicourse on total positivity ( Sergei Fomin , Euler Institute for Discrete Mathematics and its Applications)

Individual evidence

1. ↑ Mark Skandera: Introductory Notes on Total Positivity.
2. ↑ Oskar Perron: On the theory of matrices. In: Math. Ann. Volume 64, No. 2, 1907, pp. 248-263.
3. ↑ F. Gantmakher, M. Krein: Sur les matrices complètement non negatives et oscillatoires. In: Compositio Math. Volume 4, 1937, pp. 445-476.
4. ^ AM Whitney: A reduction theorem for totally positive matrices. In: J. Analyze Math. Volume 2, 1952, pp. 88-92.
5. ^ Charles Loewner: On totally positive matrices. In: Math. Z. Volume 63, 1955, pp. 338-340.
6. ^ G. Lusztig: Total positivity in reductive groups. In: Lie theory and geometry. (= Progr. Math. 123). Birkhäuser, Boston 1994, ISBN 0-8176-3761-3 , pp. 531-568.