Totally positive 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 is 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]

• 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

References

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

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