Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. The basic jamming scenario arises in weakly compressed systems of soft particles interacting through repulsive contact forces at zero temperature. In this report we propose a random matrix approach to the jamming transition. We show how to decompose a random dynamical matrix M in the form M = AA^T where A is some real rectangular random matrix. The number of rows of the matrix A is equal to the number of degrees of freedom in the system and the number of columns is equal to the number of bonds. The vibrational density of states generally depends only on the aspect ratio of the matrix A and can be described by the Debye law at small frequencies and the Marchenko-Pastur law at higher frequencies. The elastic shear modulus of the system near the jamming transition is proportional to the difference between  the numbers of columns and rows of the matrix A and equal to zero if this difference is negative. It is in agreement with numerical molecular dynamics simulation data.