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

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(\mathbf {B} )>0

for all submatrices $\mathbf {B}$ 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).