The asymptotics of eigenvalues and eigenfunctions of the Dirichlet problem for the biharmonic operator in a narrow two-dimensional domain (a thin Kirchhoff plate with rigidly clamped edges) as its width tends to zero is studied. The effect of localization of eigenfunctions is described, which consists in their exponential decay when removing away from the most wide plate region. DOI 10.1134/S1061920821020035