A matrix is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E. B. Davies: for each , every matrix is at least -close to one whose eigenvectors have condition number at worst , for some depending only on n. We further show that the dependence on δ cannot be improved to for any constant .

中文翻译：

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伪谱的高斯正则化和戴维斯猜想

如果矩阵具有线性无关的特征向量的基，则该矩阵是可对角化的。由于不可对角化矩阵集的测度为零，所以每个都是可对角化矩阵的极限。我们证明了 EB Davies 推测的这一事实的定量版本：对于每个，每个矩阵至少 -接近其特征向量在最坏情况下具有条件数的一个，对于某些仅取决于n。我们进一步表明，对于任何常数 ，对δ的依赖性都不能改善。