Totally positive matrix: Difference between revisions
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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 |
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{{hatnote|Not to be confused with [[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 [[Sign (mathematics)|positive]]. A totally positive matrix also has all positive [[eigenvalues]]. |
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==Definition== |
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Let |
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:<math>\bold{A}_{[p]} = (A_{i_kj_\ell})\,, \quad 1 \leq i_k, j_\ell \leq p \leq n </math> |
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be a ''n'' × ''n'' matrix, where ''p'' is an [[integer]] between 1 and ''n'', and |
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:<math>\begin{align} |
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& \bold{i} = (i_1, i_2,\cdots i_p) \\ |
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& \bold{j} = (j_1, j_2,\cdots j_p) |
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\end{align}</math> |
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are both [[element (mathematics)|element]]s of ℤ<sup>''p''</sup> (the ''p''-[[dimension]]al [[set (mathematics)|set]] of integers, see also [[Cartesian product]]), and let |
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:<math>A_{[p]}(\bold{i},\bold{j}) = \det(\bold{A}_{[p]})</math>. |
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Then ''A'' a '''totally positive matrix''' if:<ref>[http://www2.math.technion.ac.il/~pinkus/list.html Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus]</ref> |
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:<math>A_{[p]}(\bold{i},\bold{j}) \geq 0 </math> |
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[[Universal quantification|for all]] '''i''' and '''j''' (that is ∀ '''i''', '''j''' ∈ ℤ<sup>''p''</sup>). |
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==Examples== |
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For example, a [[Vandermonde matrix]] whose nodes are positive and increasing is a totally positive matrix. |
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==References== |
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{{reflist}} |
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==External links== |
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* [http://www2.math.technion.ac.il/~pinkus/list.html Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus] |
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* [http://www.sciencedirect.com/science/article/pii/S0001870896900572 Parametrizations of Canonical Bases and Totally Positive Matrices, Arkady Berenstein] |
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* [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] |
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{{Linear-algebra-stub}} |
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[[sl:Polno pozitivna matrika]] |
[[sl:Polno pozitivna matrika]] |
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Revision as of 13:58, 31 August 2012
In mathematics, a totally positive matrix is a square matrix in which the determinant of every square submatrix, including the minors, is positive. A totally positive matrix also has all positive eigenvalues.
Definition
Let
be a n × n matrix, where p is an integer between 1 and n, and
are both elements of ℤp (the p-dimensional set of integers, see also Cartesian product), and let
- .
Then A a totally positive matrix if:[1]
for all i and j (that is ∀ i, j ∈ ℤp).
Examples
For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.