AI 中文总结
研究方阵逆特征向量，通过留数理论框架得出谱定理逆类似物，对实相关矩阵有特殊结论，还扩展到多种矩阵及方程，论证产生有受控范数和积的逆特征向量，用于证明极化不等式等。
AI 中文摘要
我们开发了一个留数理论框架来研究由非线性方程\(M\alpha=\alpha^{-1}\)定义的方阵的逆特征向量。我们的主要结果是谱定理的逆类似物：在自然横截性和恰当性假设下，恒等算子可明确分解为与逆特征向量相关的秩一张量。证明基于多复变量有理微分形式的留数。对于非对角元素模严格小于1的实相关矩阵，我们去掉恰当性假设并表明分解中的系数为正且和为1。因此，逆特征向量支持一个明确的中心离散各向同性概率测度并形成加权紧框架。我们将构造扩展到任意实Gram矩阵、对角逆特征向量和加权逆特征向量方程。简单的迹和算术 - 几何平均论证产生具有受控欧几里得范数和坐标积的逆特征向量。这些估计导致强实极化不等式和第\(n\)个实线性极化不等式的简短证明，以及加权和矩阵值推广以及进一步的几何和分析应用。
英文摘要
We develop a residue-theoretic framework for studying inverse eigenvectors of a square matrix, defined by the nonlinear equation $Mα=α^{-1}$. Our main result is an inverse analogue of the spectral theorem: under natural transversality and properness assumptions, the identity operator admits an explicit decomposition into rank-one tensors associated with the inverse eigenvectors. The proof is based on residues of rational differential forms in several complex variables. For real correlation matrices whose off-diagonal entries have modulus strictly less than one, we remove the properness assumption and show that the coefficients in the decomposition are positive and sum to~$1$. Consequently, the inverse eigenvectors support an explicit centered discrete isotropic probability measure and form a weighted tight frame. We extend the construction to arbitrary real Gram matrices, diagonal inverse eigenvectors, and weighted inverse-eigenvector equations. Simple trace and arithmetic--geometric mean arguments yield inverse eigenvectors with controlled Euclidean norm and coordinate product. These estimates lead to short proofs of the strong real polarization inequality and the $n$th real linear polarization inequality, together with weighted and matrix-valued generalizations and further geometric and analytic applications.
Comments24 pages. First, preliminary version
