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

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]

## 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