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, 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,