totally positive matrix

Let $\backslash mathbf\{A\}\; =\; (A\_\{ij\})\_\{ij\}$

be an n × n matrix. Consider any $p\backslash in\backslash \{1,2,\backslash ldots,n\backslash \}$ and any p × p submatrix of the form $\backslash mathbf\{B\}\; =\; (A\_\{i\_kj\_\backslash ell\})\_\{k\backslash ell\}$

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:[http://www2.math.technion.ac.il/~pinkus/list.html Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus]

:$\backslash det(\backslash mathbf\{B\})\; >\; 0$

for all submatrices $\backslash mathbf\{B\}$ that can be formed this way.

Topics which historically led to the development of the theory of total positivity include the study of:

• the spectral properties of kernels and matrices which are totally positive,
• ordinary differential equations 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 properties (started by I. J. Schoenberg in 1930),
• Pólya frequency functions (by I. J. Schoenberg in the late 1940s and early 1950s).

Theorem. (Gantmacher, Krein, 1941){{Harvard citation|Fallat|Johnson|2011|p=74}} If are positive real numbers, then the Vandermonde matrix$$V\; =\; V(x\_0,\; x\_1,\; \backslash cdots,\; x\_n)\; =$$

\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} is totally positive.

More generally, let be real numbers, and let be positive real numbers, then the generalized Vandermonde matrix $V\_\{ij\}\; =\; x\_i^\{\backslash alpha\_j\}$ is totally positive.

Proof (sketch). It suffices to prove the case where $\backslash alpha\_0\; =\; 0,\; \backslash dots,\; \backslash alpha\_n\; =\; n$.

The case where are rational positive real numbers reduces to the previous case. Set $p\_i\; /\; q\_i\; =\; \backslash alpha\_i$, then let $x\text{'}\_i\; :=\; x\_i^\{1/q\_i\}$. This shows that the matrix is a minor of a larger Vandermonde matrix, so it is also totally positive.

The case where are positive real numbers reduces to the previous case by taking the limit of rational approximations.

The case where are real numbers reduces to the previous case. Let $\backslash alpha\_i\text{'}\; =\; \backslash alpha\_i\; -\; \backslash alpha\_0$, and define $V\_\{ij\}\text{'}\; =\; x\_i^\{\backslash alpha\_j\text{'}\}$. Now by the previous case, $V\text{'}$ is totally positive by noting that any minor of $V$ is the product of a diagonal matrix with positive entries, and a minor of $V\text{'}$, so its determinant is also positive.

For the case where $\backslash alpha\_0\; =\; 0,\; \backslash dots,\; \backslash alpha\_n\; =\; n$, see {{Harvard citation|Fallat|Johnson|2011|p=74}}.

• Compound matrix

{{reflist}}

• {{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.}}

• [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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