Finding normal modes of a loaded string with the method of Lagrange multipliers

We consider the normal mode problem of a vibrating string loaded with n identical beads of equal spacing, which involves an eigenvalue problem. Unlike the conventional approach to solving this problem by considering the difference equation for the components of the eigenvector, we modify the eigenvalue equation by introducing matrix-valued Lagrange undetermined multipliers, which regularize the secular equation and make the eigenvalue equation non-singular. Then, the eigenvector can be obtained from the regularized eigenvalue equation by multiplying the indeterminate eigenvalue equation by the inverse matrix. We find that the inverse matrix is nothing but the adjugate matrix of the original matrix in the secular determinant up to the determinant of the regularized matrix in the limit that the constraint equation vanishes. The components of the adjugate matrix can be represented in simple factorized forms. Finally, one can directly read off the eigenvector from the adjugate matrix. We expect this new method to be applicable to other eigenvalue problems involving more general forms of the tridiagonal matrices that appear in classical mechanics or quantum physics.