Totally positive matrix

Not to be confused with Positive matrix and Positive-definite matrix.

In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also 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 $𝐀=(A_{ij}{)}_{ij}$ be an n × n matrix. Consider any $p\in \{1,2,\dots ,n\}$ and any p × p submatrix of the form $𝐁=(A_{i_k{j}_{\ell }}{)}_{k\ell }$ where:

$1\le i_1<\dots <i_p\le n,1\le j_1<\dots <j_p\le n.$

Then A is a totally positive matrix if:

$\det (𝐁)>0$

for all submatrices $𝐁$ that can be formed this way.

History

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).

Examples

Theorem. (Gantmacher, Krein, 1941) If $0<x_0<\dots <x_n$ are positive real numbers, then the Vandermonde matrix$$V=V(x_0,x_1,\cdots ,x_n)=\left[\begin{array}{ccccc}1& x_0& x_0^2& \dots & x_0^n\\ 1& x_1& x_1^2& \dots & x_1^n\\ 1& x_2& x_2^2& \dots & x_2^n\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& x_n& x_n^2& \dots & x_n^n\end{array}\right]$$is totally positive.

More generally, let ${\alpha }_0<\dots <{\alpha }_n$ be real numbers, and let $0<x_0<\dots <x_n$ be positive real numbers, then the generalized Vandermonde matrix $V_{ij}=x_i^{{\alpha }_j}$ is totally positive.

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

The case where $0\le {\alpha }_0<\dots <{\alpha }_n$ are rational positive real numbers reduces to the previous case. Set $p_i/q_i={\alpha }_i$, 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 $0\le {\alpha }_0<\dots <{\alpha }_n$ are positive real numbers reduces to the previous case by taking the limit of rational approximations.

The case where ${\alpha }_0<\dots <{\alpha }_n$ are real numbers reduces to the previous case. Let ${{\alpha }_i}^{\prime }={\alpha }_i-{\alpha }_0$, and define $V_{i^{\prime }j}=x_i^{{{\alpha }_j}^{\prime }}$. Now by the previous case, $V^{\prime }$ is totally positive by noting that any minor of $V$ is the product of a diagonal matrix with positive entries, and a minor of $V^{\prime }$, so its determinant is also positive.

For the case where ${\alpha }_0=0,\dots ,{\alpha }_n=n$, see (Fallat & Johnson 2011, p. 74).

See also

• Compound matrix

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