Let M be the matrix representation of T with respect to the basis A and N be the matrix representation w.r.t. the basis B. To convert M to N, recall that you use
where P is the change of basis matrix for vectors expressed in the basis B to their expression in the basis A.
Let v be an eigenvector of N with eigenvalue λ. Then
Hence λ is also an eigenvalue of the matrix M for the eigenvector Pv. As this holds for any arbitrary bases A and B, we see that eigenvalues of one matrix representation are also eigenvalues of any other.