Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
This matrix equation is reminiscent of the single di€erential equation x ˆ ax, with
a a constant. The latter always has an exponential solution. This suggests that we
try
~X ˆ ~Ce !t ;
where ! is to be determined and
0
B ~C ˆ @
C 1
C 2
C 3
1
C
A
is an as yet unknown constant matrix. Substituting this into the above matrix
equation, we obtain a matrix-eigenvalue equation
~A ~C ˆ ! 2 ~C
or
0
k m
k M
B
@
0
k
m
2k
M
k
m
1
0
0
k
B
M
@
k C
A
m
C 1
C 2
C 3
1 0 1
C 1
C
A ˆ ! 2 B
@ C
C
2 A: …3:77†
C 3
Thus the possible values of ! are the square roots of the eigenvalues of the
asymmetric matrix A ~ with the corresponding solutions being the eigenvectors of
the matrix ~A. The secular equation is
k m !2
k M
0
k
m
2k
M !2
k
m
0
k
M
ˆ 0:
k m !2
This leads to
! 2 ! 2 ‡ k
! 2 ‡ k m m ‡ 2k
ˆ 0:
M
The eigenvalues are
! 2 ˆ 0;
k
m ; and k
m ‡ 2k
M ;
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