论文标题
关于Para-Hermitian矩阵的Rellich Eigendecosition和$*$ - palindromic矩阵多项式的标志特征
On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of $*$-palindromic matrix polynomials
论文作者
论文摘要
我们研究了para-Hermitian矩阵$ h（z）$的特征成分，即矩阵值函数，这些功能是分析性的，在单位圈子上是hermitian $ s^1 \ subset \ subset \ mathbb c $。特别是，我们填补了文献中的现有空白，并证明了分解$ h（z）= u（z）d（z）d（z）d（z）u（z）u（z）^p $，对于所有$ z \ in s^1 $中的所有$ z \ in s^1 $，$ u（z）$是统一的，$ u（z）^p = u（z）^p = u（z）^*$是conjugate troxpote troxpote toxpose and $ d（z）此外，对于某些正整数$ n $而言，$ u（z）$和$ d（z）$是$ W = z^{1/n} $的分析功能，而$ u（z）^p $是所谓的$ u（z）$的所谓para-hermitian conjugate。这概括了Rellich著名定理的矩阵值函数，这些功能是分析性的，而Hermitian在真实的线上。我们还表明，也存在分解$ h（z）= v（z）c（z）v（z）v（z）^p $其中$ c（z）$是伪循环，$ v（z）$是统一的，两者都是$ z $的分析。我们认为，实际上，可以将Rellich定理的版本用于矩阵值函数，该函数在复杂平面上的任何线上或任何圆上进行了分析和Hermitian。此外，我们将这些结果扩展到Para-Hermitian矩阵，其条目是Puiseux系列（也就是说，在单位圈子上，它们以$ W $分析，但可能不在$ z $中）。最后，我们讨论了结果对矩阵的奇异值分解的含义，该矩阵的条目为$ w $的$ s^1 $分析函数，以及与$*$*$*$*$*$*$*$*$ palindromic矩阵多项式的符号特征相关的符号特征。
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.
