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. 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 $$\mathbf{B} = (A_{i_kj_\ell})_{k\ell}$$ where:

1\le i_1 < \ldots < i_p \le n,\qquad 1\le j_1 <\ldots < j_p \le n. $$ Then A is a totally positive matrix if:


 * $$\det(\mathbf{B}) > 0 $$

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:


 * 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.