Required mathematical background: Matrix eigenvalues and eigenfunctions.
A study of the vibration modes of a linear triatomic molecule is used to illucidate the meaning of eigenvalues and eigenvectors in linear algebra.
Let 
, 
and 
be the masses of the atoms of a triatomic molecule.
For simplicity, the motions of the atoms are constrained to lie along the x-axis.
Their equilibrium positions are labeled by 
, 
and 
.
At time t, the positions of the atoms as measured from their respective
equilibrium positions are
, 
and 
.
Assuming that the deviations of the atoms from their equilibrium positions are small compared with the interatomic separations, the potential energy function has the following form:
where 
is the spring constant connecting atom i and
atom j.
The kinetic energy is
Newton's equations of motion
give
To further simplify the algebra, it is assumed that atoms 1 and 3 are
identical and so 
and 
.
For example, the system may represent a 
molecule.
It is clear that the time dependence of the displacements must have the form
, and so the following matrix eigenvalue
problem is obtained:
where
with 
and
.
The eigenvalue 
gives a measure of the square of the vibration
frequency.
Matrix 
is 
because there are three atoms moving along a single direction. Therefore there are 3 eigenvalues which give three normal
mode vibration frequencies. These eigenvalues are obtained from the equation
where 
is the
identity matrix.
Expanding the determinant along the first column, for example, yields the equation
which can be factored to give
The eigenvalues are 
, 
and
.
The corresponding un-normalized eigenvectors can easily be found and they are
From the eigenvector 
it is clear that the first mode corresponds
to a vibrational pattern in which the atom in the middle is stationary while
the atoms at the ends vibrate with the same amplitude but
out of phase with each other.
The second mode corresponds to a situation in which the two atoms at the ends
vibrate in phase with each other but 
out of phase with the atom in the
middle. Also the amplitude of vibration is larger for the middle atom than that
of the end atoms.
Frequency of vibration of the third mode is zero and so it is clearly not a
vibration at all. From the corresponding eigenvector, 
,
it is obvious that this mode corresponds to a rigid translation of the entire
molecule. This mode exists because there is no net force exerted on the
center-of-mass of the molecule as a whole.
A more vivid picture of the first and second mode can be obtained by exploring
the
Java applet which shows an animation of these modes.