Editing Lorentz transformation (section)

===Improper transformations===

Lorentz transformations also include [[parity inversion]]
<math display="block"> P = \begin{bmatrix} 1 & 0 \\ 0 & - \mathbf{I} \end{bmatrix} </math>
which negates all the spatial coordinates only, and [[T-symmetry|time reversal]]
<math display="block"> T = \begin{bmatrix} - 1 & 0 \\ 0 & \mathbf{I} \end{bmatrix}</math>
which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here {{math|'''I'''}} is the {{math|3 × 3}} [[identity matrix]]. These are both symmetric, they are their own inverses (see [[involution (mathematics)]]), and each have determinant −1. This latter property makes them improper transformations.

If {{math|Λ}} is a proper orthochronous Lorentz transformation, then {{math|''T''Λ}} is improper antichronous, {{math|''P''Λ}} is improper orthochronous, and {{math|1=''TP''Λ = ''PT''Λ}} is proper antichronous.