Totally positive matrix
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In mathematics, a totally positive matrix is a square matrix in which the determinant of every square submatrix, including the minors, is not negative.[1] A totally positive matrix also has all nonnegative eigenvalues.
Contents
Definition
Let
be an n × n matrix, where n, p, k, ℓ are all integers so that:
Then A a totally positive matrix if:[2]
for all p. Each integer p corresponds to a p × p submatrix of A.
History
Topics which historically led to the development of the theory of total positivity include the study of:[2]
- the spectral properties of kernels and matrices which are totally positive,
- ordinary differential equations 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 (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.
See also
References
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 2.0 2.1 Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus
Further reading
- Lua error in package.lua at line 80: module 'strict' not found.
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
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