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 be an n × n matrix. Consider any and any p × p submatrix of the form where:
Then A is a totally positive matrix if:[2]
for all submatrices 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
References
- ^ George M. Phillips (2003), "Total Positivity", Interpolation and Approximation by Polynomials, Springer, p. 274, ISBN 9780387002156
- ^ 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