Applications of Mathematics

, Volume 62, Issue 4, pp 319–331

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix



Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ Rm×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A−1) and Tr(A−2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A−1) and Tr(A−2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A−1)}2/Tr(A−2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.


eigenvalue bound symmetric positive definite matrix Laguerre bound singular value computation dqds algorithm 

MSC 2010

15A18 15A42 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.The University of Electro-CommunicationsChofu, TokyoJapan

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