Totally positive matrix
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Through Mathematics, Matrix (mathematics) and Determinant
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:
for all submatrices that can be formed this way.
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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.
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