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.

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 (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).

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

• Compound matrix

• Allan Pinkus (2009), Totally Positive Matrices, Cambridge University Press, ISBN 9780521194082

• 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