Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
Taking the dot product of Eq. (3.41) with ^e 1 0 and using the orthonormal relation
^e i 0 ^e j 0 ˆ ij (where ij is the Kronecker delta symbol), we obtain x1 0 ˆ r ^e 1.
0
Similarly, we have x2 0 ˆ r ^e 2 0 and x3 0 ˆ r ^e 3. 0 Combining these results we have
x 0
i ˆ X3
jˆ1
^e 0
i ^e j x j ˆ X3
jˆ1
ij x j ; i ˆ 1; 2; 3: …3:42†
The quantities ij ˆ ^e 0
i ^e j are called the coecients of transformation. They are
the direction cosines of the primed coordinate axes relative to the unprimed ones
or
ij ˆ ^e 0
i ^e j ˆ cos…x 0
i ; x j †; i; j ˆ 1; 2; 3: …3:42a†
Eq. (3.42) can be written conveniently in the following matrix form
0
x1
0 1 0
10
1
11 12 13 x 1
B
@ x2
0 C B
A ˆ 21 22 CB
@
23 A x C
@ 2 A
31 32 33 x 3
x 0 3
…3:43a†
~X 0 ˆ ~…† ~X; …3:43b†
where ~X 0 and ~X are the column matrices, ~ …† is called a transformation (or
rotation) matrix; it acts as a linear operator which transforms the vector X into
the vector X 0 . Strictly speaking, we should describe the matrix ~ …† as the matrix
representation of the linear operator ^. The concept of linear operator is more
general than that of matrix.
Not all of the nine quantities ij are independent; six relations exist among the
ij , hence only three of them are independent. These six relations are found by
using the fact that the magnitude of the vector must be the same in both systems:
X 3
iˆ1
…x 0
i † 2 ˆ X3
With the help of Eq. (3.42), the left hand side of the last equation becomes
! !
X 3 X 3 X 3
X 3 X 3
ij x j ik x k ij ik x j x k ;
iˆ1
jˆ1
kˆ1
iˆ1
ˆ X3
x 2 i :
iˆ1
jˆ1 kˆ1
which, by rearranging the summations, can be rewritten as
…3:44†
X 3 X 3
kˆ1 jˆ1
X 3
iˆ1
ij ik
!x j x k :
This last expression will reduce to the right hand side of Eq. (3.43) if and only if
X 3
iˆ1
ij ik ˆ jk ; j; k ˆ 1; 2; 3: …3:45†
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