A Spectral Characterization of Generalized Real Symmetric Centrosymmetric and Generalized Real Symmetric Skew-Centrosymmetric Matrices
A Spectral Characterization of Generalized Real Symmetric Centrosymmetric and Generalized Real Symmetric Skew-Centrosymmetric Matrices
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DOI：
10.1137/s0895479801386730
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发表时间：
2001-03
期刊：
影响因子：
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通讯作者：
David Tao;Mark Yasuda
中科院分区：
文献类型：
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作者：
David Tao;Mark Yasuda
证明了谱在行反转或列反转后模号不变的实对称矩阵是中心对称矩阵；此外，我们还证明了实对称中心对称矩阵可以完全用这个性质来表征。我们还证明了在行或列反转后，谱随乘以i而变化的实对称矩阵是斜中心对称矩阵；在这里，我们也证明了实对称斜中心对称矩阵可以完全用其特征值的这个性质来表征。我们证明了这两种谱表征作为我们称之为广义中心对称k矩阵和广义偏中心对称k矩阵的结果的特殊情况。给出了一些结果，说明了广义中心对称谱表征在其他实对称矩阵中的应用。
We show that the only real symmetric matrices whose spectrum is invariant modulo sign changes after either row or column reversal are the centrosymmetric matrices; moreover, we prove that the class of real symmetric centrosymmetric matrices can be completely characterized by this property. We also show that the only real symmetric matrices whose spectrum changes by multiplication by i after either row or column reversal are the skew-centrosymmetric matrices; here, too, we show that the class of real symmetric skew-centrosymmetric matrices can be completely characterized by this property of their eigenvalues. We prove both of these spectral characterizations as special cases of results for what we've called generalized centrosymmetric K-matrices and generalized skew-centrosymmetric K-matrices. Some results illustrating the application of the generalized centrosymmetric spectral characterization to other classes of real symmetric matrices are also given.