Abel polynomials: Difference between revisions

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{{hatnote|Not to be confused with [[Positive matrix]] and [[Positive-definite matrix]].}}
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In [[mathematics]], a '''totally positive matrix''' is a square [[matrix (mathematics)|matrix]] in which the [[determinant]] of every square [[submatrix]], including the [[minor (linear algebra)|minors]], is not negative.<ref>{{citation |page=274 |chapter=Total Positivity | title=Interpolation and Approximation by Polynomials |author=George M. Phillips |publisher=Springer |year=2003 |isbn=9780387002156}}</ref> A totally positive matrix also has all nonnegative [[eigenvalues]].
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==Definition==
 
Let
 
:<math>\bold{A}_{[p]} = (A_{i_kj_\ell})</math>
 
be a ''n'' × ''n'' matrix, where ''n, p, k, {{ell}}'' are all [[integer]]s so that:
 
:<math>\begin{align}
& \bold{A}_{[p]} = (A_{i_kj_\ell})\\
& {1 \leq i_k, j_\ell \leq n \text{ for }1 \leq k, \ell \leq p}
\end{align}</math>
 
Then '''A''' a '''totally positive matrix''' if:<ref name=AP>[http://www2.math.technion.ac.il/~pinkus/list.html Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus]</ref>
 
:<math>\det(\bold{A}_{[p]}) \geq 0 </math>
 
[[Universal quantification|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:<ref name=AP/>
 
* the [[Spectrum (functional analysis)|spectral]] properties of [[Kernel (algebra)|kernel]]s and matrices which are totally positive,
* [[ordinary differential equation]]s whose [[Green's function]] is totally positive (by M. G. Krein and some colleagues in the mid-1930s),
* the [[Variation diminishing property|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==
 
* [[Compound matrix]]
 
==References==
{{reflist}}
 
==Further reading==
*{{citation |title=Totally Positive Matrices |author=Allan Pinkus |publisher=[[Cambridge University Press]] |year=2009 |isbn=9780521194082}}
 
==External links==
* [http://www2.math.technion.ac.il/~pinkus/list.html Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus]
* [http://www.sciencedirect.com/science/article/pii/S0001870896900572 Parametrizations of Canonical Bases and Totally Positive Matrices, Arkady Berenstein]
* [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.3624 Tensor Product Multiplicities, Canonical Bases And Totally Positive Varieties (2001), A. Berenstein , A. Zelevinsky]
 
[[Category:Matrix theory]]
[[Category:Determinants]]
 
 
{{Linear-algebra-stub}}

Latest revision as of 18:52, 31 August 2014

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