Page:Grundgleichungen (Minkowski).djvu/27

Here I am using a method of calculation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2nd kind, and of which the rules, as far as required are given below.

1°. A system of magnitudes $a_{hk}$, formed into the matrix

$\left|{\begin{array}{ccc}a_{11},&\dots &a_{1q}\\\vdots &&\vdots \\a_{p1},&\dots &a_{pq}\end{array}}\right|$

arranged in p horizontal rows, and q vertical columns is called a $p\times q$ series-matrix,^[1] and will be denoted by the letter A.

If all the quantities $a_{hk}$ are multiplied by c, the resulting matrix will be denoted by $cA$.

If the roles of the horizontal rows and vertical columns be intercharged, we obtain a $q\times p$ series matrix, which will be known as the transposed matrix of A, and will be denoted by A.

${\bar {A}}=\left|{\begin{array}{ccc}a_{11},&\dots &a_{q1}\\\vdots &&\vdots \\a_{1p},&\dots &a_{pq}\end{array}}\right|$.

If we have a second $p\times q$ series matrix B.

$B=\left|{\begin{array}{ccc}b_{11},&\dots &b_{1q}\\\vdots &&\vdots \\b_{p1},&\dots &b_{pq}\end{array}}\right|$,

then A+B shall denote the $p\times q$ series matrix whose members are $a_{hk}+b_{hk}$.

2° If we have two matrices

$A=\left|{\begin{array}{ccc}a_{11},&\dots &a_{1q}\\\vdots &&\vdots \\a_{p1},&\dots &a_{pq}\end{array}}\right|,\ B=\left|{\begin{array}{ccc}b_{11},&\dots &b_{1r}\\\vdots &&\vdots \\b_{p1},&\dots &b_{qr}\end{array}}\right|$

where the number of horizontal rows of B, is equal to the number of vertical columns of A,

1. One could think about using Hamilton's quaternion calculus instead of Cayley's matrix calculus, however, Hamilton's calculus seems to me as too narrow and cumbersome for our purposes.