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.^[1] 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 ${\displaystyle \mathbf {A} =(A_{ij})_{ij}}$ be an n × n matrix. Consider any ${\displaystyle p\in \{1,2,\ldots ,n\}}$ and any p × p submatrix of the form ${\displaystyle \mathbf {B} =(A_{i_{k}j_{\ell }})_{k\ell }}$ where:

${\displaystyle 1\leq i_{1}<\ldots <i_{p}\leq n,\qquad 1\leq j_{1}<\ldots <j_{p}\leq n.}$

Then A is a totally positive matrix if:^[2]

${\displaystyle \det(\mathbf {B} )>0}$

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

History

Topics which historically led to the development of the theory of total positivity include the study of:^[2]

• 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)^[3] If ${\displaystyle 0<x_{0}<\dots <x_{n}}$ are positive real numbers, then the Vandermonde matrix$${\displaystyle V=V(x_{0},x_{1},\cdots ,x_{n})={\begin{bmatrix}1&x_{0}&x_{0}^{2}&\dots &x_{0}^{n}\\1&x_{1}&x_{1}^{2}&\dots &x_{1}^{n}\\1&x_{2}&x_{2}^{2}&\dots &x_{2}^{n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{n}&x_{n}^{2}&\dots &x_{n}^{n}\end{bmatrix}}}$$is totally positive.

More generally, let ${\displaystyle \alpha _{0}<\dots <\alpha _{n}}$ be real numbers, and let ${\displaystyle 0<x_{0}<\dots <x_{n}}$ be positive real numbers, then the generalized Vandermonde matrix ${\displaystyle V_{ij}=x_{i}^{\alpha _{j}}}$ is totally positive.

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

The case where ${\displaystyle 0\leq \alpha _{0}<\dots <\alpha _{n}}$ are rational positive real numbers reduces to the previous case. Set ${\displaystyle p_{i}/q_{i}=\alpha _{i}}$, then let ${\displaystyle x'_{i}:=x_{i}^{1/q_{i}}}$. This shows that the matrix is a minor of a larger Vandermonde matrix, so it is also totally positive.

The case where ${\displaystyle 0\leq \alpha _{0}<\dots <\alpha _{n}}$ are positive real numbers reduces to the previous case by taking the limit of rational approximations.

The case where ${\displaystyle \alpha _{0}<\dots <\alpha _{n}}$ are real numbers reduces to the previous case. Let ${\displaystyle \alpha _{i}'=\alpha _{i}-\alpha _{0}}$, and define ${\displaystyle V_{ij}'=x_{i}^{\alpha _{j}'}}$. Now by the previous case, ${\displaystyle V'}$ is totally positive by noting that any minor of ${\displaystyle V}$ is the product of a diagonal matrix with positive entries, and a minor of ${\displaystyle V'}$, so its determinant is also positive.

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

See also

• Compound matrix