Linear Algebra Derivation of Lorentz Transformation

Why the Lorentz transformation is linear:    • Intuition: Why the Lorentz Transforma...  

Why the eigenvalues need to be positive:
We can write the vector (β,1) as a linear combination of the two eigenvectors. If one of the eigenvalues is negative, then (β,1) will get flipped along one of the axes of the eigenvectors. Because |β| is less than 1 we know that (β,1) starts out in the top region, above y=x and y=-x. Once it gets flipped it won't be in the top region anymore. In that case, it can't possibly equal (0,1) because (0,1) is in the top region. This contradicts our assumption. Therefore the eigenvalues can't be negative because that leads to a contradiction.
The eigenvalues also can't be zero because then there would be a nonzero null space, so the map would not be injective (multiple values map to zero), so it would not have an inverse. As a result, we know that the eigenvalues must be positive.

The Lorentz transformation is often derived using thought experiments about shooting light rays and things like that. But we can prove the matrix form of the Lorentz transformation in a more abstract way using linear algebra! This gives us a way to describe changes of coordinate system when we move between inertial reference frames.

Special relativity playlist:    • Special Relativity  

0:00 Starting assumptions
6:20 Derivation

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