Mathematical Methods for Physicists: A concise introduction - Site Map

DIRECT PRODUCT OF MATRICES
Figure 3.6.
Longitudinal vibrations of a carbon dioxide molecule.
all real. The corresponding eigenvectors are determined by substituting the eigenvalues
back into Eq. (3.77) one eigenvalue at a time:
(1) Setting ! 2 ˆ 0 in Eq. (3.77) we ®nd that C 1 ˆ C 2 ˆ C 3 . Thus this mode is
not an oscillation at all, but is a pure translation of the system as a whole, no
relative motion of the masses (Fig. 3.6(a)).
(2) Setting ! 2 ˆ k=m in Eq. (3.77), we ®nd C 2 ˆ 0 and C 3 ˆC 1 . Thus the
center mass M is stationary while the outer masses vibrate in opposite
directions with the same amplitude (Fig. 3.6(b)).
(3) Setting ! 2 ˆ k=m ‡ 2k=M in Eq. (3.77), we ®nd C 1 ˆ C 3 , and
C 2 ˆ2C 1 …m=M†. In this mode the two outer masses vibrate in
unison and the center mass vibrates oppositely with di€erent amplitude
(Fig. 3.6(c)).
Direct product of matrices
Sometimes the direct product of matrices is useful. Given an m m matrix ~A
and an n n matrix ~B, the direct product of A ~ and ~B is an mn mn matrix,
de®ned by
0
1
a 11
~B a 12
~B a 1m
~B
a 21
~B a 22
~B a 2m
~B
~C ˆ ~A ~B ˆ
.
.
. . :
B
C
@
A
a m1
~B a m2
~B a mm
~B
For example, if
!
!
~A ˆ a11 a 12
; ~B ˆ b11 b 12
;
a 21 a 22 b 21 b 22
then
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