Biomedical Engineering Reference
In-Depth Information
Fig. A.1
The relative rotational orientation between coordinate systems
2
3
10 0
0
4
5
:
Q
¼
cos
y
sin
y
(A.65)
0
sin
y
cos
y
This situation is illustrated on the left in Fig.
A.1
.
The matrix
Q
[
Q
i
a
] characterizing the change from the Latin orthonormal
basis
e
i
in an N-dimensional vector space to the Greek basis
e
a
(or vice versa) is a
special type of linear transformation called an orthogonal transformation. Taking
the scalar product of
e
i
with
e
j
where
e
i
and
e
j
both have the representation (A.62),
¼
e
i
¼
Q
i
a
e
a
and
e
j
¼
Q
j
b
e
b
;
(A.66)
it follows that
e
i
e
j
¼ d
ij
¼
Q
i
a
Q
j
b
e
a
e
b
¼
Q
i
a
Q
j
b
d
ab
¼
Q
i
a
Q
j
a
:
(A.67)
There are a number of steps in the calculation (A.67) that should be considered
carefully. First, the condition of orthonormality of the bases has been used twice,
e
i
e
b
¼ d
ab
. Second, the transition from the term before the last
equal sign to the term after that sign is characterized by a change from a double sum
to a single sum over
n
and the loss of the Kronecker delta
e
j
¼ d
ij
and
e
a
d
ab
. This occurs because
the sum over
b
in the double sum is always zero except in the special case when
a ¼ b
d
ab
. Third, a comparison of the
last term in (A.67) with the definition of matrix product (A.20) suggests that it is a
matrix product of
Q
with itself. However, a careful comparison of the last term in
(A.67) with the definition of matrix product (A.20) shows that the summation is
over a different index in the second element of the product. In order for the last term
in (A.67) to represent a matrix product, the
due to the presence of the Kronecker delta
index should appear as the first
subscripted index rather than the second. However, this
a
index may be relocated
in the second matrix by using the transposition operation. Thus the last term in
equation (A.67) is the matrix product of
Q
with
Q
T
as may be seen from the first of
a
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