_i.e._ the reciprocal matrix of A is equivalent to the transposed matrix
of A.
For A as Lorentz transformation, we have further Det A = +1, the
quantities involving the index 4 once in the subscript are purely
imaginary, the other co-efficients are real, and _a₄₄_ > 0.
5⁰. A space time vector of the first kind[21] which s represented by the
1 × 4 series matrix,
(41) _s_ = |_s₁_ _s₂_ _s₃_ _s₄_|
is to be replaced by _s_A in case of a Lorentz transformation
A. _i.e._ _s′_ = | _s₁′_ _s₂′_ _s₃′_ _s₄′_| = |_s₁_ _s₂_ _s₃_ _s₄_|
A;
A space-time vector of the 2nd kind[22] with components _f₂₃_ ... _f₃₄_
shall be represented by the alternating matrix
(42) _f_ = | 0 _f_₁₂ _f₁₃_ _f₁₄_ |
|_f₂₁_ 0 _f_₂₃ _f₂₄_ |
|_f_₃₁ _f₃₂_ 0 _f₃₄_ |
|_f_₄₁ _f_₄₂ _f_₄₃ 0 |
and is to be replaced by A⁻¹ _f_ A in case of a Lorentz transformation
[see the rules in § 5 (23) (24)]. Therefore referring to the expression
(37), we have the identity Det^{½} (Ā _f_ A) = Det A. Det^{½} _f_.
Therefore Det^{½} _f_ becomes an invariant in the case of a Lorentz
transformation [see eq. (26) See. § 5].
Looking back to (36), we have for the dual matrix (Ā_f_*A) (A⁻¹_f_A) =
A⁻¹_f_*_f_A = Det^{½} function. A⁻¹A = Det^{½}_f_ from which it is to be
seen that the dual matrix _f_* behaves exactly like the primary matrix
_f_, and is therefore a space time vector of the II kind; _f_* is
therefore known as the dual space-time vector of _f_ with components
(_f₁₄_, _f₂₄_, _f₃₄_,), (_f₂₃_}, _f₃₁_, _f₁₂_).
6. If _w_ and _s_ are two space-time rectors of the 1st kind then by _w_
_ṡ_ (as well as by _s_ _ẇ_) will be understood the combination (43) _w₁_
_s₁_ + _w₂_ _s₂_ + _w₃_ _s₃_ + _w₄_ _s₄_.
In case of a Lorentz transformation A, since (_w_A) (Ā_ṡ_) = _w_ _s_,
this expression is invariant.—If _w_ _ṡ_ = 0, then _w_ and _s_ are
perpendicular to each other.
Two space-time rectors of the first kind (_w_, _s_) gives us a 2 × 4
series matrix
| _w₁_ _w₂_ _w₃_ _w₄_ |
| _s₁_ _s₂_ _s₃_ _s₄_ |