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. A totally positive matrix also has all nonnegative eigenvalues.

Definition

Let

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

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

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

Then A is a totally positive matrix if:

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

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:

• 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).

Examples

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

See also

• Compound matrix

Further reading

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

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