On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of *-palindromic matrix polynomials
On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of *-palindromic matrix polynomials
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准Hermitian矩阵的Rellich特征分解及*-回文矩阵多项式的符号特征
DOI：
10.48550/arxiv.2211.15539
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
V. Noferini
中科院分区：
文献类型：
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作者：
Giovanni Barbarino;V. Noferini
研究单位圆$S^1\子集\MathbbC$上的仿厄米特矩阵H(Z)$的特征分解，即解析的、厄米特的矩阵值函数.特别地，我们填补了文献中的空白，证明了分解$H(Z)=U(Z)D(Z)U(Z)^P$的存在性，其中对于S^1$中的所有$z，$U(Z)$是酉解，$U(Z)^P=U(Z)^*$是它的共轭转置，$D(Z)$是实对角线的；此外，$U(Z)$和$D(Z)$是$N$的$w=z^{1/N}$的解析函数，$U(Z)^P$是$U(Z)$的所谓仿厄米特共轭.这推广了著名的Rellich定理，它适用于在实线上是解析的和厄米特的矩阵值函数。我们还证明了也存在分解$H(Z)=V(Z)C(Z)V(Z)^P$，其中$C(Z)$是伪循环的，$V(Z)$是酉解的，两者都是在$z$中解析的。我们认为，实际上，对于任意直线或复平面上任意圆上的解析式和厄米特式的矩阵值函数，Rellich定理的一种形式可以表述。此外，我们将这些结果推广到输入为Puiseux级数的仿厄米特矩阵(即在单位圆上它们是$w$解析的，但可能不是$z$解析的)。最后，我们讨论了我们的结果对元素为$w$的$S^1$解析函数的矩阵的奇异值分解以及与$*$-回文矩阵多项式的么模特征值有关的符号特征的影响。
We study the eigendecompositions of para-Hermitian matrices $H(z)$, that is, matrix-valued functions that are analytic and Hermitian on the unit circle $S^1 \subset \mathbb C$. In particular, we fill existing gaps in the literature and prove the existence of a decomposition $H(z)=U(z)D(z)U(z)^P$ where, for all $z \in S^1$, $U(z)$ is unitary, $U(z)^P=U(z)^*$ is its conjugate transpose, and $D(z)$ is real diagonal; moreover, $U(z)$ and $D(z)$ are analytic functions of $w=z^{1/N}$ for some positive integer $N$, and $U(z)^P$ is the so-called para-Hermitian conjugate of $U(z)$. This generalizes the celebrated theorem of Rellich for matrix-valued functions that are analytic and Hermitian on the real line. We also show that there also exists a decomposition $H(z)=V(z)C(z)V(z)^P$ where $C(z)$ is pseudo-circulant, $V(z)$ is unitary and both are analytic in $z$. We argue that, in fact, a version of Rellich's theorem can be stated for matrix-valued function that are analytic and Hermitian on any line or any circle on the complex plane. Moreover, we extend these results to para-Hermitian matrices whose entries are Puiseux series (that is, on the unit circle they are analytic in $w$ but possibly not in $z$). Finally, we discuss the implications of our results on the singular value decomposition of a matrix whose entries are $S^1$-analytic functions of $w$, and on the sign characteristics associated with unimodular eigenvalues of $*$-palindromic matrix polynomials.