Positive semidefinite univariate matrix polynomials
Positive semidefinite univariate matrix polynomials
复制标题
正半定单变量矩阵多项式
DOI：
10.1007/s00209-018-2137-7
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发表时间：
2017
影响因子：
0.8
通讯作者：
Rainer Sinn
中科院分区：
文献类型：
--
作者：
Christoph Hanselka;Rainer Sinn
我们研究沿实数线半正定的对称单变量实数矩阵多项式的平方和表示。我们给出了一个新的证明，即每个大小的正半定单变量矩阵多项式都可以写成平方和，其中 Q 具有大小，最近由 Blekherman-Plaumann-Sinn-Vinzant 证明了这一点。我们使用二次形式理论的新方法使我们能够证明这些作者的猜想，即这些最小表示通常与非负单变量多项式的两个平方和的表示一一对应。同时，我们将使用我们的方法来证明更基本的埃尔米特类似物，即沿实数直线为正半定的每个埃尔米特单变量矩阵多项式 M 都是一个正方形，这称为矩阵 Fejér–Riesz 定理。
We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of sizecan be written as a sum of squares, whereQhas size, which was recently proved by Blekherman–Plaumann–Sinn–Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representationsare generically in one-to-one correspondence with the representations of the nonnegative univariate polynomialas sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomialMthat is positive semidefinite along the real line, is a square, which is known as the matrix Fejér–Riesz Theorem.