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.^[1] 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 ${\displaystyle \mathbf {A} =(A_{ij})_{ij}}$ be an n × n matrix. Consider any ${\displaystyle p\in \{1,2,\ldots ,n\}}$ and any p × p submatrix of the form ${\displaystyle \mathbf {B} =(A_{i_{k}j_{\ell }})_{k\ell }}$ where:

${\displaystyle 1\leq i_{1}<\ldots <i_{p}\leq n,\qquad 1\leq j_{1}<\ldots <j_{p}\leq n.}$

Then A is a totally positive matrix if:^[2]

${\displaystyle \det(\mathbf {B} )>0}$

for all submatrices ${\displaystyle \mathbf {B} }$ that can be formed this way.

History

Topics which historically led to the development of the theory of total positivity include the study of:^[2]

• 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