DEFLATION FOR THE SYMMETRIC ARROW MATRIX EIGENVALUE PROBLEM JESSE L. BARLOW * , NEVENA JAKOV ˇ CEVI ´ C STOR, AND IVAN SLAPNI ˇ CAR † Abstract. We discuss the eigenproblem for the symmetric arrow matrix C = „ D z z T α « where D ∈ R n×n is diagonal, z ∈ R n , and α ∈ R. New criteria for when components of z may be set to zero are proposed. We show that whenever two eigenvalues of C are clustered near a diagonal, without significantly perturbing the eigenvalues of C, the corresponding component of z may be set to zero either by substituting zero for that component or perform a Givens rotation on each side of C. That means that if C has a cluster of k eigenvalues, k - 1 of them can be deflated. The resulting deflation algorithm can be used to develop a more aggressive method for finding converged eigenvalues in the symmetric Lanczos algorithm. The algorithm is extended to the singular value decomposition of the half-arrow matrix to include a deflation approach that preserves the singular values of the half-arrow matrix to relative accuracy. AMS subject classifications. 65F15, 65G50, 15-04,15B99 Key words. Symmetric arrow matrix, half-arrow matrix, eigenvalue and singular value defla- tion, Krylov-Schur, symmetric Lanczos algorithm. 1. Introduction. For a given integer n, consider the eigenvalue problem for the symmetric arrow matrix C ∈ R (n+1)×(n+1) given by C =  n 1 n D z 1 z T α  (1.1) where D ∈ R n×n is a diagonal matrix, z ∈ R n , and α ∈ R. Without loss of generality, let D = diag(d 1 ,d 2 ,...,d n ) (1.2) where d 1 ≥ d 2 ≥ ... ≥ d n . (1.3) Symmetric arrow matrices arise in the description of radiationless transitions in isolated molecules [4], oscillators vibrationally coupled with a Fermi liquid [7], and quantum optics [16]. Such matrices also arise in solving symmetric real tridiago- nal eigenvalue problems with the divide-and-conquer method [10], in updating the symmetric eigenvalue problem [8], and in determining eigenvalue convergence in the symmetric Lanczos algorithm [8]. The singular value version—the half-arrow singular * THE RESEARCH OF JESSE L. BARLOW WAS SPONSORED BY THE NATIONAL SCI- ENCE FOUNDATION UNDER CONTRACT NO. CCF-1115704. † THE RESEARCH OF NEVENA JAKOV ˇ CEVI ´ C STOR AND IVAN SLAPNI ˇ CAR WAS SPON- SORED BY THE CROATIAN SCIENCE FOUNDATION UNDER CONTRACT NO. IP-2014009- 9540. 1