The Number of Distinct Eigenvalues of a Matrix After Perturbation
The Number of Distinct Eigenvalues of a Matrix After Perturbation
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DOI：
10.1137/15m1037603
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发表时间：
2015-08
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影响因子：
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通讯作者：
P. Farrell
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文献类型：
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作者：
P. Farrell
我们证明了一个新的定理有关的数目不同的特征值的矩阵扰动后的先验数目不同的特征值，更新的秩，和程度的非对角化的矩阵。特别地，应用于可对角化矩阵的秩一更新最多可以使不同特征值的数量加倍。该定理适用于对称和非对称矩阵和扰动，任意大小。作为一个应用，我们证明了在精确算术中，精确求解一个包含可对角化矩阵的线性系统所需的Krylov迭代次数在秩一更新后最多可以加倍。
We prove a new theorem relating the number of distinct eigenvalues of a matrix after perturbation to the prior number of distinct eigenvalues, the rank of the update, and the degree of nondiagonalizability of the matrix. In particular, a rank one update applied to a diagonalizable matrix can at most double the number of distinct eigenvalues. The theorem applies to both symmetric and nonsymmetric matrices and perturbations, of arbitrary magnitudes. An an application, we prove that in exact arithmetic the number of Krylov iterations required to exactly solve a linear system involving a diagonalizable matrix can at most double after a rank one update.