Mathematical Methods for Physicists: A concise introduction - Site Map

TRACE OF A MATRIX
that is, by , we recover the original coordinate system. Thus
0
1 0
1
0
B
~R…† ~R…† ˆ@
0 1
C
0A ˆ ~R 1 …† ~R…†:
0 0 1
Hence
0
1
cos sin 0
~R 1 B
C
…† ˆ ~R…† ˆ@
sin cos 0 A ˆ ~R T …†;
0 0 1
which shows that a rotation matrix is an orthogonal matrix.
We would like to make one remark on rotation in space. In the above discussion,
we have considered the vector to be ®xed and rotated the coordinate axes.
The rotation matrix can be thought of as an operator that, acting on the unprimed
system, transforms it into the primed system. This view is often called the passive
view of rotation. We could equally well keep the coordinate axes ®xed and rotate
the vector through an equal angle, but in the opposite direction. Then the rotation
matrix would be thought of as an operator acting on the vector, say X, and
changing it into X 0 . This procedure is called the active view of the rotation.
Trace of a matrix
Recall that the trace of a square matrix A ~ is de®ned as the sum of all the principal
diagonal elements:
Tr ~A ˆ X
a kk :
It can be proved that the trace of the product of a ®nite number of matrices is
invariant under any cyclic permutation of the matrices. We leave this as home
work.
k
Orthogonal and unitary transformations
Eq. (3.42) is a linear transformation and it is called an orthogonal transformation,
because the rotation matrix is an orthogonal matrix. One of the properties of an
orthogonal transformation is that it preserves the length of a vector. A more
useful linear transformation in physics is the unitary transformation:
~Y ˆ ~U ~X …3:48†
in which ~X and ~Y are column matrices (vectors) of order n 1 and ~U is a unitary
matrix of order n n. One of the properties of a unitary transformation is that it
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