More uses for eigenvalues and eigenvectors

It turns out that looking at eigenvalues and eigenvectors may be useful for more than thinking about linear recursions like the Fibonnaci sequence. They're also be useful when thinking about special relativity. In this case we have that the coordinates (x',t') in a frame of reference moving at velocity v relative to a stationary frame of reference are related to the coordinates (x,t) in the stationary frame of reference by

 
x'
t'
 
= γ
 
1 v
v 1
 
 
x
t
 

where γ = 1 / √(1 – v2) and speed of light is normalized to c = 1.

Now the matrix

γ
 
1 v
v 1
 

has eigenvectors

 

 
-1
1
 
and
 
1
1
 

which correspond to the eigenvalues γ(1 + v) and γ(1 - v) respectively. One interpretation for this is that the natural coordinates system to use has  

 
-1
1
 
and
 
1
1
 

for its basis instead of

 
1
0
 
and
 
0
1
 

If we had use those coordinates, lots of physics would be much simpler. On the other hand, that would also make it more difficult to actually make any measurements. Maybe our current system isn't really so bad after all.

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