| α₄₁, α₄₂, α₄₃, α₄₄ |
will be denoted as the transformation A.
By the transformation A, the expression
x²₁ + x²₂ + x²₃ + x²₄ is changed into the quadratic for m ∑ αhk xh′ xk′,
where αhk = α1k α1k + α2h α2k + α3h α3k + α4h α4k are the members of a 4 × 4 series matrix which is the product of Ā A, the transposed matrix of A into A. If by the transformation, the expression is changed to
x′₁² + x₂′2 + x₃′2 + x′₄²,
we must have Ā A = 1.
A has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of A) it follows out of (39) that (Det A)² = 1, or Det A = ± 1.
From the condition (39) we obtain
A⁻¹ = Ā,