# 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

$\bold{A} = (A_{ij})$

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

\begin{align} & \bold{A}_{[p]} = (A_{i_kj_\ell})\\ & {1 \leq i_k, j_\ell \leq n \text{ for }1 \leq k, \ell \leq p} \end{align}

Then A a totally positive matrix if:[2]

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

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.

## References

1. ^ George M. Phillips (2003), "Total Positivity", Interpolation and Approximation by Polynomials, Springer, p. 274, ISBN 9780387002156
2. ^ a b Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus