View source for Totally positive matrix

{{hatnote|Not to be confused with [[Positive matrix]] and [[Positive-definite matrix]].}}

In [[mathematics]], a '''totally positive matrix''' is a square [[matrix (mathematics)|matrix]] in which all the [[minor (linear algebra)|minors]] are positive: that is, the [[determinant]] of every square [[submatrix]] is a positive number.<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 has all entries positive, so it is also a [[positive matrix]]; and it has all [[Minor (linear algebra)#Other applications|principal minor]]s positive (and positive [[eigenvalue]]s). A [[symmetric matrix|symmetric]] totally positive matrix is therefore also [[positive-definite matrix|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 <math>\mathbf{A} = (A_{ij})_{ij}</math>
be an ''n'' × ''n'' matrix. Consider any <math>p\in\{1,2,\ldots,n\}</math> and any ''p'' × ''p'' submatrix of the form <math>\mathbf{B} = (A_{i_kj_\ell})_{k\ell}</math>
where:
:<math>
1\le i_1 < \ldots < i_p \le n,\qquad 1\le j_1 <\ldots <  j_p \le n.
</math>
Then '''A''' is 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(\mathbf{B}) > 0 </math>

for all submatrices <math>\mathbf{B}</math> that can be formed this way.

==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 [[Positive-definite kernel|kernels]] and matrices which are totally positive,
* [[ordinary differential equation]]s whose [[Green's function]] is totally positive, which arises in the theory of mechanical vibrations (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==

Theorem. (Gantmacher, Krein, 1941)<ref>{{Harvard citation|Fallat|Johnson|2011|p=74}}</ref> If <math>0 < x_0 < \dots < x_n</math> are positive real numbers, then the [[Vandermonde matrix]]<math display="block">V = V(x_0, x_1, \cdots, x_n) =
\begin{bmatrix}
1 & x_0 & x_0^2 & \dots & x_0^n\\
1 & x_1 & x_1^2 & \dots & x_1^n\\
1 & x_2 & x_2^2 & \dots & x_2^n\\
\vdots & \vdots & \vdots & \ddots &\vdots \\
1 & x_n & x_n^2 & \dots & x_n^n
\end{bmatrix} </math>is totally positive. 

More generally, let <math>\alpha_0 < \dots < \alpha_n</math> be real numbers, and let <math>0 < x_0 < \dots < x_n</math> be positive real numbers, then the generalized Vandermonde matrix <math>V_{ij} = x_i^{\alpha_j}</math> is totally positive.

'''Proof''' (sketch). It suffices to prove the case where <math>\alpha_0 = 0, \dots, \alpha_n = n</math>.

The case where <math>0 \leq \alpha_0 < \dots < \alpha_n</math> are rational positive real numbers reduces to the previous case. Set <math>p_i / q_i = \alpha_i</math>, then let <math>x'_i := x_i^{1/q_i}</math>. This shows that the matrix is a minor of a larger Vandermonde matrix, so it is also totally positive.

The case where <math>0 \leq \alpha_0 < \dots < \alpha_n</math> are positive real numbers reduces to the previous case by taking the limit of rational approximations.

The case where <math>\alpha_0 < \dots < \alpha_n</math> are real numbers reduces to the previous case. Let <math>\alpha_i' = \alpha_i - \alpha_0</math>, and define <math>V_{ij}' = x_i^{\alpha_j'}</math>. Now by the previous case, <math>V'</math> is totally positive by noting that any minor of <math>V</math> is the product of a diagonal matrix with positive entries, and a minor of <math>V'</math>, so its determinant is also positive.

For the case where <math>\alpha_0 = 0, \dots, \alpha_n = n</math>, see {{Harvard citation|Fallat|Johnson|2011|p=74}}.

==See also==

* [[Compound matrix]]

==References==
{{reflist}}

==Further reading==
*{{citation |title=Totally Positive Matrices |author=Allan Pinkus |publisher=[[Cambridge University Press]] |year=2009 |isbn=9780521194082}}
*{{Cite book |title=Totally nonnegative matrices |date=2011 |publisher=Princeton University Press |isbn=978-0-691-12157-4 |editor-last=Fallat |editor-first=Shaun M. |series=Princeton series in applied mathematics |location=Princeton |editor-last2=Johnson |editor-first2=Charles R.}}

==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]

{{Matrix classes}}

[[Category:Matrix theory]]
[[Category:Determinants]]


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