Page 517 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

504 Determinants and Matrices Appen. C

Partitioned Matrices
A m atrix can be partitioned into subm atrices by horizontal and vertical lines, as
shown by the following example:
'2 4 ! - r
0 - 3 1 4
1 2 1 2
J _ [C ] [ D ]
_3 - 1 1 - 5 ,
where the subm atrices are
4
A = - 3
2
C = [3 - 1 ]
Partitioned m atrices obey the norm al rules of m atrix algebra and can be added,
subtracted, and m ultiplied as though the subm atrices were ordinary m atrix ele­
ments. Thus,
A 1 B /5 .\ A { x ] + B [ y )
c \ D . C { x ) + D { y )
A 1 B E 1 F A E + B G \ A F + B H
c ‘ D g " \ H . C E + D G \ C F + D H

C.4 DETERMINATION OF EIGENVECTORS

The eigenvector corresponding to the eigenvalue A- can be found from the
cofactors of any row of the characteristic equation.
Let [ A - A J ] X ^ = 0 be written out for a third-order system as
a
(^11 13
(^22 ^23 = 0 (C .4 -1)
^32 33 ~ ^ i )
Its characteristic equation |^ — Ay/| = 0 written out in determ inant form is
( « 11- A,) «12 ^13
(^22 ^23 = 0 (C .4 -2)
^31 ^32 (^33 ^i )
The determ inant expanded in terms of the cofactors of the first row is
(«11 — A ,)C ii ^12^12 ^13^13 ^ ^ (C .4 -3)