Improving the condition number of a simple eigenvalue by a rank one matrix

In this work a technique to improve the condition number sisi of a simple eigenvalue λiλi of a matrix A∈Cn×nA∈Cn×n is given. This technique obtains a rank one updated matrix that is similar to AA with the eigenvalue condition number of λiλi equal to one. More precisely, the similar updated matrix A+viq∗A+viq∗, where Avi=λiviAvi=λivi and qq is a fixed vector, has si=1si=1 and the remaining condition numbers are at most equal to the corresponding initial condition numbers. Moreover an expression to compute the vector qq, using only the eigenvalue λiλi and its eigenvector vivi, is given.