Mmat={{(m1+m2)L1^2,m2 L1 L2},{m2 L1 L2,m2 L2^2}}

Kmat=DiagonalMatrix[{(m1+m2) g L1,m2 g L2}]

FREQ=Simplify[Eigenvalues[Inverse[Mmat].Kmat]]

Simplify[FREQ/.{m1->m,m2->m,L1->L,L2->L},{L>0,m>0}]

FREQ=Simplify[Eigenvectors[Inverse[Mmat].Kmat]/.{m1->m,m2->m,L1->L,L2->L},{L>0,m>0}]

Mmat3=m L^2 DiagonalMatrix[{1,1,1}];Kmat3=m g L DiagonalMatrix[{1,1,1}]+k L^2 {{1,-1,0},{-1,2,-1},{0,-1,1}};

Eigenvalues[Inverse[Mmat3].Kmat3]

Eigenvectors[Inverse[Mmat3].Kmat3]