Totally positive matrix explained

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. 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_ij)_ij

be an n × n matrix. Consider any

p\in\{1,2,\ldots,n\}

and any p × p submatrix of the form

(A
	i_kj_\ell

)_k\ell

where:

1\lei₁<\ldots<i_p\len, 1\lej₁<\ldots<j_p\len.

Then A is a totally positive matrix if:^[1]

\det(B)>0

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:^[1]

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.

External links

Notes and References

http://www2.math.technion.ac.il/~pinkus/list.html Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus