Totally positive matrix

Definition

Let \mathbf{A} = (A_{ij}){ij} be an n × n matrix. Consider any p\in{1,2,\ldots,n} and any p × p submatrix of the form \mathbf{B} = (A{i_kj_\ell})_{k\ell} where: : 1\le i_1 Then A is a totally positive matrix if:

:\det(\mathbf{B}) 0

for all submatrices \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:

• the spectral properties of kernels and matrices which are totally positive,
• ordinary differential equations whose Green's function is totally positive, which arises in the theory of mechanical vibrations (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

Theorem. (Gantmacher, Krein, 1941) If 0 are positive real numbers, then the Vandermonde matrixV = V(x_0, x_1, \cdots, x_n) = \begin{bmatrix} 1 & x_0 & x_0^2 & \dots & x_0^n\ 1 & x_1 & x_1^2 & \dots & x_1^n\ 1 & x_2 & x_2^2 & \dots & x_2^n\ \vdots & \vdots & \vdots & \ddots &\vdots \ 1 & x_n & x_n^2 & \dots & x_n^n \end{bmatrix} is totally positive.

More generally, let \alpha_0 be real numbers, and let 0 be positive real numbers, then the generalized Vandermonde matrix V_{ij} = x_i^{\alpha_j} is totally positive.

Proof (sketch). It suffices to prove the case where \alpha_0 = 0, \dots, \alpha_n = n.

The case where 0 \leq \alpha_0 are rational positive real numbers reduces to the previous case. Set p_i / q_i = \alpha_i, then let x'_i := x_i^{1/q_i}. This shows that the matrix is a minor of a larger Vandermonde matrix, so it is also totally positive.

The case where 0 \leq \alpha_0 are positive real numbers reduces to the previous case by taking the limit of rational approximations.

The case where \alpha_0 are real numbers reduces to the previous case. Let \alpha_i' = \alpha_i - \alpha_0, and define V_{ij}' = x_i^{\alpha_j'}. Now by the previous case, V' is totally positive by noting that any minor of V is the product of a diagonal matrix with positive entries, and a minor of V', so its determinant is also positive.

For the case where \alpha_0 = 0, \dots, \alpha_n = n, see .

References

References

1. George M. Phillips. (2003). "Interpolation and Approximation by Polynomials". Springer.
2. [http://www2.math.technion.ac.il/~pinkus/list.html Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus]
3. {{Harvard citation. Fallat. Johnson. 2011