  In this work, we investigate a symmetric deformed random matrix, which is
obtained by perturbing the diagonal elements of the Wigner matrix. The
eigenvector $\mathbf{x}_{\rm min}$ of the minimal eigenvalue $\lambda_{\rm
min}$ of the deformed random matrix tends to condensate at a single site. In
certain types of perturbations and in the limit of the large components, this
condensation becomes a sharp phase transition, the mechanism of which can be
identified with the Bose-Einstein condensation in a mathematical level. We
study this Bose-Einstein like condensation phenomenon by means of the replica
method. We first derive a formula to calculate the minimal eigenvalue and the
statistical properties of $\mathbf{x}_{\rm min}$. Then, we apply the formula
for two solvable cases: when the distribution of the perturbation has the
double peak, and when it has a continuous distribution. For the double peak, we
find that at the transition point, the participation ratio changes
discontinuously from a finite value to zero. On the contrary, in the case of a
continuous distribution, the participation ratio goes to zero either
continuously or discontinuously, depending on the distribution.
