The Eigenvalue Problem For Symmetric Matrices

Matrix eigenvalue problems arise, for example, from Hamilton’s principle; the latter states: A mechanical system whose kinetic and potential energy are given by(1)$$T = \sum\limits_{i = 1}^n {} \sum\limits_{j = 1}^n {} {P_i}_j({q_1},...,{q_n}){\dot q_i}{\dot q_j},\,\,\,\,U = U({q_1},...,{q_n})$$, evolves between the time instances t₀ and t₁ in such a way that the functions q _i (t) describing the motion make the action integral$$J = \int_{{t_0}}^{{t_1}} {} (T - U)dt$$stationary, the values q _i (t ₀ ) 2nd qi(t₁) being held fixed.