On Pre-Conditioning of Matrices
On Pre-Conditioning of Matrices
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关于矩阵的预处理
DOI：
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发表时间：
1960
期刊：
影响因子：
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通讯作者：
E. E. Osborne
中科院分区：
文献类型：
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作者：
E. E. Osborne
在获得矩阵的ElgenValues和特征向量的问题中遇到的一些困难似乎是由于其特征值与其规范相比很小。第3节中提供的示例倾向于验证该声明。在第2节中，简要考虑将规范相似性转换应用于A的可能性，从而决定将转换矩阵限制为对角线。为了证明这一点是合理的，在第3节中给出了示例。这些示例表明涉及选择关键元素的浮点矩阵计算，并且可以从内部产物的形成中受益。对角矩阵的这种转换还提供了一种用于缩放定点计算矩阵的方法。在第4节中，介绍了迭代过程，并证明了其收敛性。在接下来的内容中，让A成为具有复杂元素的序列矩阵，并且具有特征值H〜（a）（i = 1，2，。，。，n）。符号II X II用于表示欧几里得规范
Some of the difficulties encountered in the problem of obtaining the elgenvalues and eigenvectors of a matrix appear to be due to the fact tha t its eigenvalues are small compared to its norm. Examples provided in section 3 tend to verify this statement. In section 2 the possibility of applying norm-reducing similarity transformations to A is briefly considered, leading to a decision to restrict the transforming matrices to being diagonal. To justify this, examples are given in section 3. These indicate tha t floating-point matrix computations involving the selection of pivotal elements and the formation of inner products may be benefited. Such transformations by diagonal matrices also provide a means for scaling a matrix for fixed-point computations. In section 4, an iterative process is presented and its convergence is proved. In what follows, let A be an n th order matrix with complex elements and with eigenvalues h~(A) (i = 1, 2, . . , n) . The symbol II x II is used to denote the Euclidean norm