Page 748 - Mechanical Engineers' Handbook (Volume 2)

4 Solution of System Equations 739

Table 4 (Continued)
VI. Nondiagonal Jordan canonical form
Transformation conditions
(i) F matrix has one repeated, real eigenvalue z k of multiplicity m (i.e., z k z k 1
z k m 1 ). All other eigenvalues are real and distinct.
(ii) Degeneracy d n rank(z k I F) 1. Simple degeneracy.
Transformation matrices
(i) q Tx, T 1 [t 1 t 2 .. . t n ]
where (a) t i , i 1, k 1 and i k m, n, are the linearly independent eigenvectors
corresponding to the real, distinct eigenvalues;
(b) t i , for i 1, k 1 and i k m, n, are taken to be equal or proportional to
any nonzero column of Adj(s i I F); and
1
(c) t i , i k, k m 1, are obtained by solution of the equation FT 1 T J,
where J is the Jordan canonical matrix given. Each t i is determined to within a
constant of proportionality.
(ii) New state matrix J TFT 1

z 1 0

0 z k 1 0 0

z 1 0
k
0 1 0
0 z k

0
z k m 1
0 0
0 z n
VII. Nondiagonal Jordan canonical form
Transformation conditions
(i) F matrix has one repeated, real eigenvalue of multiplicity m (i.e., z k z k 1
z k m 1 ). All other eigenvalues are real and distinct.
(ii) Degeneracy d n rank(z k I F) is between 1 and m. Simple degeneracy.
Transformation matrices
(i) q Tx, T 1 [t 1 t 2 .. . t n ]
where (a) t i , i 1, k 1 and i k m, n, are the linearly independent eigenvectors
corresponding to the real, distinct eigenvalues;
(b) t i , i 1, k 1 and i k m, n, are taken to be equal or proportional to any
nonzero column of Adj(s i I F);
(c) t i , i k, k m 1, are m linearly independent vectors corresponding to the
repeated eigenvalue of multiplicity m, only d of these vectors eigenvectors; and
1
(d) t i , i k k, k m 1, obtained by solution of the equation FT 1 T J,
where J is the Jordan canonical matrix for the problem, with d Jordan blocks.
There are d possible choices for J. Each of these choices needs to be tried and
the t i vectors solved for. Only the correct J will give the m linearly
independent vectors t i , i k, k m 1. Each t i is determined to within a
constant of proportionality.
1
(ii) New state matrix J TFT . Correct J determined by trial and error, as previously
described.