Page 266 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.10 Jacobi Diagonalization 253

8.10 JACOBI DIAGONALIZATION
In the section on orthogonality, Sec. 6.7 in Chapter 6, the assembling of the
orthonormal eigenvectors cf) into the modal matrix P enabled the mass and the
stiffness matrices to be expressed in the basic relationships:

P^MP = I ( 8.10-1)
P^KP = A

where / is a unit matrix, and A is a diagonal matrix of the eigenvalues. These
relationships indicate that if the eigenvectors of the system are known, the
eigenvalue problem is solved.
The Jacobi method is based on the principle that any real symmetric matrix
A has only real eigenvalues and can be diagonalized into the eigenvalue matrix
A = tA,J by an iteration method. In the Jacobi method, this is accomplished by
several rotation matrices R by which the off-diagonal elements of A are zeroed by
repeated iterations until matrix A is diagonalized. The method is developed for
the standard eigenproblem equation:
{A - k I ) Y = 0 ( 8.10-2)

and the major advantage of the procedure is that all of the eigenvalues and
eigenvectors are found simultaneously.
In the standard eigenproblem, the M and K matrices have already been
transposed into a single symmetric dynamic matrix A, which is more economical
for iteration than two matrices. The kth iteration step is defined by the equations

~ ^k+\
(8.10-3)

^k+\^k+l^k+\ ^k +2 , etc.
where R,^ is the rotation matrix.
Before discussing the general problem of diagonalizing the dynamic matrix A
of nth order, it will be helpful to demonstrate the Jacobi procedure with an
elementary problem of a second-order matrix:

A =
^22
The rotation matrix for this case is simply the orthogonal matrix
cos 0 —sin 0

R = (8.10-4)
sin 6 cos 0