..
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?

ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>[[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]]

Thanks for that observation!

In the meantime, I found the answer to my question reading a text
by Viktor T. Toth.

Many Textbooks say,

( -1 0 0 0 )
eta_{mu nu} = ( 0 1 0 0 )
( 0 0 1 0 )
( 0 0 0 1 ),

but when you multiply this by a column (contravariant) vector,
you get another column (contravariant) vector instead of a row,
while the "v_mu" in

eta_{mu nu} v^nu = v_mu

seems to indicate that you will get a row (covariant) vector!

As Viktor T. Toth observed in 2005, a square matrix (i.e.,
a row of columns) only really makes sense for eta^mu_nu (which is
just the identity matrix). He then clear-sightedly explains that
a matrix with /two/ covariant indices needs to be written not
as a /row of columns/ but as a /row of rows/:

eta_{mu nu} = [( -1 0 0 0 )( 0 1 0 0 )( 0 0 1 0 )( 0 0 0 1 )]
. Now, if one multiplies /this/ with a column (contravariant)
vector, one gets a row (covariant) vector (tweaking the rules for
matrix multiplication a bit by using scalar multiplication for
the product of the row ( -1 0 0 0 ) with the first row of the
column vector [which first row is a single value] and so on)!

Exercise Work out the representation of eta^{mu nu} in the same
spirit.