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From: ram@zedat.fu-berlin.de (Stefan Ram)
Newsgroups: sci.physics.research
Subject: Matrix Multiplication in SR
Date: Tue, 30 Jul 2024 23:44:26 PDT
Organization: Stefan Ram
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..
  I have read the following derivation in a chapter on SR.

|(0) We define:
|X := p_"mu" p^"mu",
|
|(1) from this, by Eq. 2.36 we get:
|= p_"mu" "eta"^("mu""nu") p_"mu",

[[Mod. note -- I think that last subscript "mu" should be a "nu".
That is, equations (0) and (1) should read (switching to LaTeX notation)
$X := p_\mu p^\mu
   =  p_\mu \eta^{\mu\nu} p_\nu$
-- jt]]

|
|(2) from this, using matrix notation, we get:
|
|                       (  1  0  0  0 ) ( p_0 )
|= ( p_0 p_1 p_2 p_3 )  (  0 -1  0  0 ) ( p_1 )
|                       (  0  0 -1  0 ) ( p_2 )
|                       (  0  0  0 -1 ) ( p_3 ),
|
|(3) from this, we get:
|= p_0 p_0 - p_1 p_1 - p_2 p_2 - p_3 p_3,
|
|(4) using p_1 p_1 - p_2 p_2 - p_3 p_3 =: p^"3-vector" * p^"3-vector":
|= p_0 p_0 - p^"3-vector" * p^"3-vector".

  . Now, I used to believe that a vector with an upper index is
  a contravariant vector written as a column and a vector with
  a lower index is covariant and written as a row. We thus can
  write (0) in two-dimensional notation:
  
                       ( p^0 )
= ( p_0 p_1 p_2 p_3 )  ( p^1 )
                       ( p^2 )
                       ( p^3 )
                       
  So, I have a question about the transition from (1) to (2):

  In (1), the initial and the final "p" both have a /lower/ index "mu".
  In (2), the initial p is written as a row vector, while the final p
  now is written as a column vector.

  When, in (1), both "p" are written exactly the same way, by what
  reason then is the first "p" in (2) written as a /row/ vector and
  the second "p" a /column/ vector? 

  Let's write p_"mu" "eta"^("mu""nu") p_"mu" with two row vectors,
  as it should be written:
  
                       (  1  0  0  0 )
= ( p_0 p_1 p_2 p_3 )  (  0 -1  0  0 )  ( p_0 p_1 p_2 p_3 )
                       (  0  0 -1  0 )
                       (  0  0  0 -1 )

  . AFAIK, the laws for matrix multiplication just do not define
  a product of a 4x4 matrix with a 1x4 matrix, because for every
  row of the left matrix, there has to be a whole column of the
  right matrix of the same size. Does this show there's something
  off with that step of the calculation?