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Totally positive matrix

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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.

Contents

Definition

Let

A = ( A i j )

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

A [ p ] = ( A i k j ℓ ) 1 ≤ i k , j ℓ ≤ n  for  1 ≤ k , ℓ ≤ p

Then A is a totally positive matrix if:

det ( A [ p ] ) ≥ 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.

    References

    Totally positive matrix Wikipedia
    (Text) CC BY-SA