# Typical Mathematical Questions about the Lorentz-Transformation

In SR (Special Relativity) we use the Lorentztransformation (LT) to relate one frame of reference to another. Say S to S' via a LT that depends only on v where v is the relative velocity between S and S'.

In Mathematics (and QM) we seem to be always interessted in the Kernel, the Range and the Eigenvalues and vectors of a matrix.

What do these things mean with respect to the LT?

After some thinking I concluded: The Kernel represents stuff that also moves with v relative to S or stuff that rests in S'. The Range should be anything that does not move with v.

I have no Idea what the Eigenvalues and Eigenvectors do.

Am I right so far? What is the meaning of the eigenstuff?

• Lorenz transformations are invertible (and they must be, since the inverse transforms the frame back), hence the kernel is always trivial and the range is always the whole space. – Martin May 6 '15 at 13:30

• Eigenvalues: how much a particular characteristic direction is strethed or shrunk.. Transformations can leave directions invariant while still dilating or shrinking the vectors along those directions by some factor. However, the Lorentz transformation is orthogonal, preserving all inner products (spacetime intervals). That includes the magnitude of any vector, so even the vectors along characteristic directions retain the same magnitude as before. That means the eigenvalues associated with those directions are $+1$. In the plane of the boost, there are no real eigenvectors (and therefore no associated real eigenvalues), but some authors might consider "complex eigenvalues" or "complex eigenvectors". I personally prefer instead to consider "eigenplanes" or such instead, which would still have completely real eigenvalues.