An example of importance of off-diagonal error matrix elements
can be found from high energy physics in the decay of a spin 1
particle.
One can define the angular distribution by density matrix
elements
in some kinematic coordinate system (Lorentz frame)!
Some years ago, these were always given in one
frame---the so-called t channel frame.
``Recently'' (1975---these really are old notes!), it was found
that the values in another frame (the ``s channel'') were more
meaningful physically.
Unfortunately, latter values, y, were related by precisely such
a linear transformation y=Tx and one can no longer estimate errors
in y from recorded experimental data.
