Totally positive matrix

Not to be confused with Positive matrix and Positive-definite 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 [math]\displaystyle{ \mathbf{A} = (A_{ij})_{ij} }[/math] be an n × n matrix. Consider any [math]\displaystyle{ p\in\{1,2,\ldots,n\} }[/math] and any p × p submatrix of the form [math]\displaystyle{ \mathbf{B} = (A_{i_kj_\ell})_{k\ell} }[/math] where:

[math]\displaystyle{ 1\le i_1 \lt \ldots \lt i_p \le n,\qquad 1\le j_1 \lt \ldots \lt j_p \le n. }[/math]

Then A is a totally positive matrix if:

[math]\displaystyle{ \det(\mathbf{B}) \gt 0 }[/math]

for all submatrices [math]\displaystyle{ \mathbf{B} }[/math] 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.

See also

• Compound matrix

Further reading

• Allan Pinkus (2009), Totally Positive Matrices, Cambridge University Press, ISBN 9780521194082 

External links

• Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus
• Parametrizations of Canonical Bases and Totally Positive Matrices, Arkady Berenstein
• Tensor Product Multiplicities, Canonical Bases And Totally Positive Varieties (2001), A. Berenstein, A. Zelevinsky