Totally positive matrix

Not to be confused with Positive-definite matrix.

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

${\mathbf {A}}_{[p]}=(A_{i_{k}j_{\ell }})\,,\quad 1\leq i_{k},j_{\ell }\leq p\leq n$

be a n × n matrix, where p is an integer between 1 and n, and

${\begin{aligned}&{\mathbf {i}}=(i_{1},i_{2},\cdots i_{p})\\&{\mathbf {j}}=(j_{1},j_{2},\cdots j_{p})\end{aligned}}$

are both elements of ℤ^p (the p-dimensional set of integers, see also Cartesian product), and let

$A_{[p]}({\mathbf {i}},{\mathbf {j}})=\det({\mathbf {A}}_{[p]})$.

Then A a totally positive matrix if:

$A_{[p]}({\mathbf {i}},{\mathbf {j}})\geq 0$

for all i and j (that is ∀ i, j ∈ ℤ^p).

Examples

For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive 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