Abstract   In this work a technique to improve the condition number     s    i     of a simple eigenvalue     λ    i     of a matrix   A  ∈    C    n  ×  n     is given. This technique obtains a rank one updated matrix that is similar to   A   with the eigenvalue condition number of     λ    i     equal to one. More precisely, the similar updated matrix   A  +    v    i      q    ∗    , where   A    v    i    =    λ    i      v    i     and   q   is a fixed vector, has     s    i    =  1   and the remaining condition numbers are at most equal to the corresponding initial condition numbers. Moreover an expression to compute the vector   q  , using only the eigenvalue     λ    i     and its eigenvector     v    i    , is given.
['Rafael Bru', 'Rafael Cantó', 'Ana M. Urbano']
Improving the condition number of a simple eigenvalue by a rank one matrix