# 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 the determinant of every square submatrix, including the minors, is not negative.[1] A totally positive matrix also has all nonnegative eigenvalues.

## Definition

Let

${\displaystyle {\mathbf {A}}=(A_{ij})}$

be an n × n matrix, where n, p, k, ℓ are all integers so that:

{\displaystyle {\begin{aligned}&{\mathbf {A}}_{[p]}=(A_{i_{k}j_{\ell }})\\&{1\leq i_{k},j_{\ell }\leq n{\text{ for }}1\leq k,\ell \leq p}\end{aligned}}}

Then A a totally positive matrix if:[2]

${\displaystyle \det({\mathbf {A}}_{[p]})\geq 0}$

for all p. Each integer p corresponds to a p × p submatrix of A.

## History

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

## Examples

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