Page:The principle of relativity (1920).djvu/105

 will be known as the transposed matrix of A, and will be denoted by Ā.

Ā = | a_{1 1} a_{p 1} | |                            |       | a_{1 q}   a_{p q} |

If we have a second p × q series matrix B,

B = | b_{1 1} b_{1 q} | |                                |    | b_{p 1}      b_{p q} |

then A + B shall denote the p × q series matrix whose members are a_{h k} + b_{h k}.

2^o If we have two matrices

A= | a_{1 1} a_{1 q} | B = | b_{1 1}  b_{1 r} | |                           |     |                          |   | a_{p 1}   a_{p q} |     | b_{q 1}  b_{p r} |

where the number of horizontal rows of B, is equal to the number of vertical columns of A, then by AB, the product of the matrics A and B, will be denoted the matrix

C = | c_{1 1} c_{1 r} | |                          |    | c_{p r}  c_{p p} |

where c_{h k} = a_{h 1} b_{1 k} + a_{h 2} b_{2 h} + a_{k s} b_{s k} +  + a_{k q} b_{q h}

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 = BA, we have C̄ = B̄Ā