Rank two perturbations of matrices and operators and operator model for t-transformation of probability measures

Rank two parametric perturbations of operators and matrices are studied in various settings. In the finite dimensional case the formula for a characteristic polynomial is derived and the large parameter asymptotics of the spectrum is computed. The large parameter asymptotics of a rank one perturbation of singular values and condition number are discussed as well. In the operator case the formula for a rank two transformation of the spectral measure is derived and it appears to be the t-transformation of a probability measure, studied previously in the free probability context. New transformation of measures is studied and several examples are presented.