18 2Basics
r 1 ′=T 11 r 1 +T 12 r 2 +T 13 r 3. (2.20)More general, forμ=1, 2, 3, one has
rμ′=Tμ 1 r 1 +Tμ 2 r 2 +Tμ 3 r 3 , (2.21)or, with the help of the summation convention
rμ′=Tμνrν. (2.22)Notice:in(2.22),μis afree indexwhich can have any value 1, 2 or 3. The subscript
ν, on the other hand, is a summation index, for which any other Greek letter, except
μ, could be chosen here.
Sometimes, the relations (2.19) or equivalently (2.22) are expressed in the form
r′=T·r, (2.23)where the matrix-character ofTis indicated by thebold face sans serifletter and
the center dot “·” implies the summation of products of components.
Notice: in such a notation, the order of factors matters, in contradistinction to the
component notation. The equationr′=r·Tcorresponds torμ′ =rνTνμ=Tνμrν
which is different from (2.22), unless the transformation matrixTis symmetric, i.e.
unlessTνμ=Tμνholds true.
Theinverse transformation, also calledback-transformation, links the compo-
nents ofrwith those ofr′, according to
r=T−^1 ·r′, (2.24)with the inverse transformation matrixT−^1. Insertion of (2.23)into(2.24) leads to
r=T−^1 ·r′=T−^1 ·T·rwhich implies
T−^1 ·T=δ, (2.25)or in component notation,
Tμλ−^1 Tλν=δμν. (2.26)
Notice: hereμandνare free indices,λis the summation index. The symbolδ
indicates the unit matrix, viz.:
δ:=⎛
⎝
100
010
001
⎞
⎠, (2.27)
or equivalently,δμν=1forμ=ν, andδμν=0forμ=ν.