Page:Grundgleichungen (Minkowski).djvu/30

 where

$$a_{hk} = \alpha_{1h}\alpha_{1k} + \alpha_{2h}\alpha_{2k} + \alpha_{3h}\alpha_{3k} + \alpha_{4h}\alpha_{4k}$$

are the members of a 4✕4 series matrix which is the product of $$\mathsf{\bar{A}A}$$, the transposed matrix of $$\mathsf{A}$$ into $$\mathsf{A}$$. If by the transformation, the expression is changed to

$$x^{'2}_{1} + x^{'2}_{2} + x^{'2}_{3} + x^{'2}_{4}$$

we must have

$$\mathsf{A}$$ has to correspond to the following relation, if transformation (38) is to be a -transformation. For the determinant of $$\mathsf{A}$$ it follows out of (39) that $$(Det \mathsf{A})^{2} = 1, Det \mathsf{A} = \pm 1$$.

From the condition (39) we obtain

i.e. the reciprocal matrix of $$\mathsf{A}$$ is equivalent to the transposed matrix of $$\mathsf{A}$$.

For $$\mathsf{A}$$ as transformation, we have further $$Det \mathsf{A} = + 1$$, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and $$\alpha_{44}>0$$.

5°. A space time vector of the first kind which is represented by the 1✕4 series matrix,

is to be replaced by $$s\mathsf{A}$$ in case of a transformation

A space-time vector of the 2nd kind with components $$f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$$ shall be represented by the alternating matrix

and is to be replaced by $$\mathsf{\overline{A}}f\mathsf{A}=\mathsf{A}^{-1}f\mathsf{A}$$ in case of a transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression