This paper is concerned with the perturbation of a multiple eigenvalue μ of the Hermitian matrix A = diag(μI, A22) when it undergoes an off-diagonal Email addresses: email@example.com (Ren-Cang Li), firstname.lastname@example.org (Yuji Nakatsukasa), email@example.com (Ninoslav Truhar), firstname.lastname@example.org (Wei-guo Wang) Supported in part by National Science Foundation Grants DMS-0810506 and DMS1115817. Supported in part by Engineering and Physical Sciences Research Council grant EP/I005293/1. Supported in part by the “Passive control of mechanical models ”, Grant Nr. 2352352818-1042 of the Croatian MZOS Supported in part by the National Natural Science Foundation of China under grant 10971204 and 11071228, China Scholarship Council, Shandong Province Natural Science Foundation (Y2008A07). This work was initiated while this author was visiting University of Texas at Arlington from September 2010 to August 2011. Preprint submitted to Linear Algebra and its Applications January 24, 2012 perturbation E whose columns have widely varying magnitudes. When some of E’s columns are much smaller than the others, some copies of μ are much less sensitive than any existing bound suggests. We explain this phenomenon by establishing individual perturbation bounds for different copies of μ. They show that when A22−μI is definite the ith bound scales quadratically with the norm of the ith column, and in the indefinite case the bound is necessarily proportional to the product of E’s ith column norm and E’s norm. An extension to the generalized Hermitian eigenvalue problem is also presented.