Upper bounds for singular values

Let A be an n × n matrix with singular values σ1 ⩾ ⋯ ⩾ σn. If 1 ⩽ r ⩽ n, then σr=minH∈Sr‖H‖, where Sr is the set of n × n matrices H such that rank(A + H) ⩽ r − 1 and ∥·∥ denotes the spectral norm, i.e., the largest singular value. We find upper bounds for σr by choosing H suitably.