Matrix Formulation
- Page ID
- 2248
Matrix representation of the equations
The matrix representation of the above equations for the kinetic and potential energies is:
where F is the force constant matrix in internal coordinates, G-1 is the transformed inverse mass matrix, and S is a vector of internal coordinates. The normal coordinates are linearly related to the internal coordinates by
S = LQ
In which the transformation coefficients are chosen so that the energies in terms of the normal coordinates have the diagonal forms
Where L is a diagonal matrix whose elements are lk = 4p2n2 and E is the unit matrix. Therefore
The second equation implies that LT = L-1G which can be applied to the left-hand side of the first equation to yield
GFL = LL
which when multiplied on both the right and the left by L-1 gives
L-1GF = L L-1
whose transpose is
FG(L-1)T = (L-1)TL
The condition of compatibility is
|GF - Elk| = 0
This formulation of harmonic analysis problem is the standard matrix formulation.