Totally positive matrix

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 A = ( A i j ) i j {\displaystyle \mathbf {A} =(A_{ij})_{ij}} be an n × n matrix. Consider any p ∈ { 1 , 2 , … , n } {\displaystyle p\in \{1,2,\ldots ,n\}} and any p × p submatrix of the form B = ( A i k j ℓ ) k ℓ {\displaystyle \mathbf {B} =(A_{i_{k}j_{\ell }})_{k\ell }} where:

1 ≤ i 1 < … < i p ≤ n , 1 ≤ j 1 < … < j p ≤ n . {\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]

det ( B ) > 0 {\displaystyle \det(\mathbf {B} )>0}

for all submatrices B {\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]

  • 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. ^ a b 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
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