Totally positive matrix

Not to be confused with Positive matrix and Positive-definite 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

{\mathbf {A}}=(A_{ij})

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

{\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]

\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

↑ George M. Phillips (2003), "Total Positivity", Interpolation and Approximation by Polynomials, Springer, p. 274, ISBN 9780387002156
1 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

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