Piecewise Certificates of Positivity for matrix polynomials

Details Piecewise Certificates of Positivity for matrix polynomials Piecewise Certificates of Positivity for matrix polynomials

2010-01-08

arxiv.org/abs/1001.1277v1

Mathematics

Rings and Algebras

Scientific paper

We show that any symmetric positive definite homogeneous matrix polynomial $M\in\R[x_1,...,x_n]^{m\times m}$ admits a piecewise semi-certificate, i.e. a collection of identites $M(x)=\sum_jf_{i,j}(x)U_{i,j}(x)^TU_{i,j}(x)$ where $U_{i,j}(x)$ is a matrix polynomial and $f_{i,j}(x)$ is a non negative polynomial on a semi-algebraic subset $S_i$, where $\R^n=\cup_{i=1}^r S_i$. This result generalizes to the setting of biforms. Some examples of certificates are given and among others, we study a variation around the Choi counterexample of a positive semi-definite biquadratic form which is not a sum of squares. As a byproduct we give a representation of the famous non negative sum of squares polynomial $x^4z^2+z^4y^2+y^4x^2-3 x^2y^2z^2$ as the determinant of a positive semi-definite quadratic matrix polynomial.

Affiliated with

Also associated with

No associations

Rating

Piecewise Certificates of Positivity for matrix polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Piecewise Certificates of Positivity for matrix polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Piecewise Certificates of Positivity for matrix polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-502511