Newton’s equation of motion for a rigid body states:
For a structure, the resultant of inertial forces are balanced by the resultant of elastic deformation forces. The equation of motion for a structure is:
(1)
where M is the structure dynamical matrix and K is the structure stiffness matrix.
Equation 1 has a solution of the type:
Substitute for x in Equation 1:
Arrange in standard eignenvalue form:
(2)
or
(3)
where Equation 2 is for M non-singular while Equation 3 is for K semi-definite.
Put in standard eigenvalue form:
Results in:
(4)
is in general not symmetric even though K and M are
symmetric.
Let 
where L is the Cholesky factor of M.
Replace M in Equation 3 with its Cholesky factors:
Let
Whence:
(5)
Equation 5 is the symmetric form of Equation 4.
Equation 5 can now be solved for eigenvalues
The fundamental frequency of vibration of the structure is given by:
