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A Constrained Eigenvalue Problem

  • Walter Gander
  • Gene H. Golub
  • Urs von Matt
Part of the NATO ASI Series book series (volume 70)

Abstract

In this paper we consider the following mathematical and computational problem. Given the quantities
  • A: (n + m)-by-(n + m) matrix, symmetric, n > 0

  • N: (n + m)-by-m matrix with full rank

  • t: vector of dimension m with ∥(NT)+t∥ < 1

Determine an x such that
$$ {\operatorname{x} ^T}Ax = \min $$
subject to the constraints
$$ {N^T}x = t $$
(i)
$$ {x^T}x = 1. $$
(ii)
Variants of this problem occur in many applications [1,5,7,8,11]. The problem has been studied previously when t = 0, the null vector, (cf. [4,6]). When t ≠ 0, then the problem becomes more complicated.

We show how to eliminate the linear constraint (i). Then three different methods are presented for the solution of the resulting Lagrange equations.

Keywords

Eigenvalue Problem Linear Constraint Full Rank Lagrange Equation Null Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Walter Gander
    • 1
  • Gene H. Golub
    • 2
  • Urs von Matt
    • 1
  1. 1.Institut für InformatikETH ZentrumZürichSwitzerland
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA

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