Page:Grundgleichungen (Minkowski).djvu/28

 then by AB, the product of the matrices A and B, will be denoted the matrix

$$C=\left|\begin{array}{ccc} c_{11}, & \dots & c_{1r}\\ \vdots & & \vdots\\ c_{p1}, & \dots & c_{pr}\end{array}\right|$$

where

$$c_{hk}=a_{h1}b_{1k}+a_{h2}b_{2k}+\dots+a_{hq}b_{qk}\quad\left({h=1,2,\dots p\atop k=1,2,\dots r}\right)$$

these elements being formed by combination of the horizontal rows of A with the vertical columns of B. For such a point, the associative law $$(AB)S = A(BS)$$ holds, where S is a third matrix which has got as many horizontal rows as B (or AB) has got vertical columns.

For the transposed matrix of $$ C = AB$$, we have $$\bar{C}=\bar{B}\bar{A}$$.

3°. We shall have principally to deal with matrices with at most four vertical columns and for horizontal rows.

As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 ✕ 4 series) with the elements.

For a 4✕4 series-matrix, Det A shall denote the determinant formed of the 4✕4 elements of the matrix. If $$Det A \ne 0$$, then corresponding to A there is a reciprocal matrix, which we may denote by $$A^{-1}$$ so that $$A^{-1} A = 1$$

A matrix

$$f=\left|\begin{array}{cccc} 0, & f_{12}, & f_{13}, & f_{14}\\ f_{21}, & 0, & f_{23}, & f_{24}\\ f_{31}, & f_{32}, & 0, & f_{34}\\ f_{41}, & f_{42}, & f_{43}, & 0\end{array}\right|$$,