Equality conditions for lower bounds on the smallest singular value of a bidiagonal matrix

Several lower bounds have been proposed for the smallest singular value of a square matrix, such as Johnson's bound, Brauer-type bound, Li's bound and Ostrowskitype bound. In this paper, we focus on a bidiagonal matrix and investigate the equality conditions for these bounds. We show that the former three bounds give strictly lower bounds if all the bidiagonal elements are nonzero. For the Ostrowskitype bound, we give an easily veri able necessary and su cient condition for the equality to hold. Key words: Singular values, Lower bounds, Equality conditions, Bidiagonal matrix, dqds algorithm.

• Type: Journal
• Year: 2008
• Where: AMC
• Authors: Yusaku Yamamoto
• Keywords: AMC 2007, AMC 2008, Lower Bounds, Several Lower Bounds, Singular Values