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 formula_1
be an "n" × "n" matrix. Consider any formula_2 and any "p" × "p" submatrix of the form formula_3
where:
Then A is a totally positive matrix if:
for all submatrices formula_6 that can be formed this way.
History.
Topics which historically led to the development of the theory of total positivity include the study of:
Examples.
For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.