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,
i.e. the reciprocal matrix of A is equivalent to the transposed matrix of A.
For A as Lorentz transformation, we have further Det A = +1, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and a_{44} > 0.
5^o. A space time vector of the first kind[1] which s
represented by the 1 × 4 series matrix,
(41) s = |s_{1} s_{2} s_{3} s_{4}|
is to be replaced by sA in case of a Lorentz transformation
A. i.e. s´ = | s_{1}´ s_{2}´ s_{3}´ s_{4}´| = |s_{1} s_{2} s_{3} s_{4}| A;
A space-time vector of the 2nd kind[2] with components f_{2 3} . . . f_{34} shall be represented by the alternating matrix
(42) f = | 0 f_{12} f_{13} f_{14} |
|f_{21} 0 f_{23} f_{24} |
|f_{31} f_{32} 0 f_{34} |
|f_{41} f_{42} f_{43} 0 |
and is to be replaced by A^{-1} f A in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression (37), we have the identity
Det^{1/2} (Ā f A) = Det A. Det^{1/2} f. Therefore Det^{1/2} f becomes an invariant in the case of a Lorentz transformation [see eq. (26) See. § 5].
