Plz help me with this linear algebra proof problem Thank you

Plz help me with this linear algebra proof problem. Thank you so much!!!

Suppose that {v_1 rightarrow, ..., v_n rightarrow} is a basis of R^n which are eigenvectors of both A and B. That is, say (lambda_1, v_1 rightarrow), .... (lambda_n, v_n rightarrow) are eigenpairs of A (rho_1, v_1 rightarrow), ... (rho_n, v_n rightarrow) are eigenpairs of B. Show that AB = BA.

Solution

Let P be the matrix with v₁ , v₂, …,v_n as its columns and let D and S be diagonal matrices with the corresponding eigenvalues of A and B respectively as the entries on the respective leading diagonals. Then D and S are diagonal matrices of the same size (nxn) so that DS = SD.

Further, since A = PDP^-1 and B = PSP^-1, we have AB = (PDP^-1)( PSP^-1) = PD(P^-1P)SP^-1 = PDI_n SP^-1 = PDSP^-1.

Also, BA = ( PSP^-1) (PDP^-1) = PS(P^-1P)DP^-1 = PSI_nDP^-1 = PSDP^-1.

Now, since DS = SD, we have AB = BA.