Total positivity

In mathematics, a totally positive matrix is a square matrix in which the determinant of every square submatrix, including the minors, is not negative. A totally positive matrix also has all nonnegative eigenvalues.

Let

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

Then A is a totally positive matrix if:

for all p. Each integer p corresponds to a p × p submatrix of A.

Topics which historically led to the development of the theory of total positivity include the study of:

For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.

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