In 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'' (franchi ...

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 ...

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 minors 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 denot ...

s). A

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

totally positive matrix is therefore also positive-definite. 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 properties of

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...

s 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 contras ...

s whose

Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...

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

Examples

For example, a

Vandermonde matrix In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \dots & x_2^\\ 1 & x ...

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