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 $𝐀=(A_{ij}{)}_{ij}$ be an n × n matrix. Consider any $p\in \{1,2,\dots ,n\}$ and any p × p submatrix of the form $𝐁=(A_{i_k{j}_{\ell }}{)}_{k\ell }$ where:

$1\le i_1<\dots <i_p\le n,1\le j_1<\dots <j_p\le n.$

Then A is a totally positive matrix if:

$\det (𝐁)>0$

for all submatrices $𝐁$ 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.

See also

• Compound matrix

Further reading

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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

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