Page:The principle of relativity (1920).djvu/106

 3^o. 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.

(34) | e_{1 1} e_{1 2} e_{1 3} e_{1 4} | = | 1 0 0 0 | | e_{2 1} e_{2 2} e_{2 3} e_{2 4} |  | 0 1 0 0 | | e_{3 1} e_{3 2} e_{3 3} e_{3 4} |  | 0 0 1 0 | | e_{4 1} e_{4 2} e_{4 3} e_{4 4} |  | 0 0 0 1 |

For a 4 × 4 series-matrix, Det A shall denote the determinant formed of the 4 × 4 elements of the matrix. If det A [symbol]?] 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 = | 0      f_{1 2} f_{1 3} f_{1 4} | | f_{2 1} 0      f_{2 3} f_{2 4} | | f_{3 1} f_{3 2} 0      f_{3 4} | | f_{4 1} f_{4 2} f_{4 3} 0      |

in which the elements fulfil the relation f_{h k} = -f_{h k}, is called an alternating matrix. These relations say that the transposed matrix [=f] = -f. Then by f^{*} will be the dual, alternating matrix

(35)

f^{*} = | 0      f_{3 4} f_{4 2} f_{2 3} | | f_{4 3} 0      f_{1 4} f_{3 1} | | f_{2 4} f_{4 1} 0      f_{1 2} | | f_{3 2} f_{1 3} f_{2 1} 0      |