mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a totally positive matrix is a square

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

in which all the minors are positive: that is, the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

of every square submatrix is a positive number. A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all

principal minor In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors ...

s positive (and positive

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s). A symmetric totally positive matrix is therefore also

positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...

. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero). Some authors use "totally positive" to include all totally non-negative matrices.

Definition

Let

\mathbf = (A_)_

be an ''n'' × ''n'' matrix. Consider any

p\in\

and any ''p'' × ''p'' submatrix of the form

\mathbf = (A_)_

where: :

1\le i_1 < \ldots < i_p \le n,\qquad 1\le j_1 <\ldots <  j_p \le n.

Then A is a totally positive matrix if:Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus
/ref> :

\det(\mathbf) > 0

for all submatrices

\mathbf

that can be formed this way.

History

Topics which historically led to the development of the theory of total positivity include the study of: * the

spectral ''Spectral'' is a 2016 3D military science fiction, supernatural horror fantasy and action-adventure thriller war film directed by Nic Mathieu. Written by himself, Ian Fried, and George Nolfi from a story by Fried and Mathieu. The film stars J ...

properties of kernels and matrices which are totally positive, *

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

s whose Green's function is totally positive (by M. G. Krein and some colleagues in the mid-1930s), * the variation diminishing properties (started by I. J. Schoenberg in 1930), *

Pólya frequency functions Pólya (Hungarian for "swaddling clothes") is a surname. People with the surname include: * Eugen Alexander Pólya (1876-1944), Hungarian surgeon, elder brother of George Pólya ** Reichel-Polya Operation, a type of partial gastrectomy developed ...

(by I. J. Schoenberg in the late 1940s and early 1950s).

Examples

For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.

References

External links

Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus

Parametrizations of Canonical Bases and Totally Positive Matrices, Arkady Berenstein

Tensor Product Multiplicities, Canonical Bases And Totally Positive Varieties (2001), A. Berenstein , A. Zelevinsky
Matrix theory Determinants {{Linear-algebra-stub

Definition

History

Examples

See also

References

Further reading

External links