Totally positive matrix

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Not to be confused with Positive matrix and Positive-definite matrix.

In mathematics, a totally positive matrix is a square matrix in which the determinant of every square submatrix, including the minors, is not negative.[1] A totally positive matrix also has all nonnegative eigenvalues.



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

Then A a totally positive matrix if:[2]

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:[2]


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

See also[edit]


  1. ^ George M. Phillips (2003), "Total Positivity", Interpolation and Approximation by Polynomials, Springer, p. 274, ISBN 9780387002156 
  2. ^ a b Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus

Further reading[edit]

External links[edit]