Biomedical Engineering Reference
In-Depth Information
2
3
123
245
356
4
5
G
¼
are 11.345, 0.171, and
0.516.
.
ab
bc
A.5.2. Construct the inverse of the matrix
A
where
A
¼
A.5.3. Show that the inverse of the matrix
G
of problem A.5.1 is given by
2
3
1
32
G
1
4
5
:
¼
33
1
2
10
A.5.4. Show that the eigenvalues of the matrix
G
-1
of problem A.5.3 are the
inverse of the eigenvalues of the matrix
G
of problem A.5.1.
A.5.5. Solve the matrix equation
A
2
¼ AA ¼ A
for
A
assuming that
A
is non-
singular.
A.5.6. Why is it not possible to construct the inverse of an open product matrix,
a
b
?
A.5.7. Construct a compositional transformation based on the matrix
G
of problem
A.5.1 and the open product matrix,
a
b
, where the n-tuples are
a
¼
[1, 2, 3]
and
b
¼
[4, 5, 6].
If
F
is a square matrix and
a
is an n-tuple, show that
a
T
F
T
A.5.8.
¼
F
a
.
A.6 Vector Spaces
Loosely, vectors are defined as
n
-
tuples
that follow the
parallelogram law of
addition
. More precisely vectors are defined as elements of a vector space called
the
arithmetic n
-
space
. Let
A
n
denote the set of all n-tuples,
u
¼
[
u
1
,
u
2
,
u
3
,
...
,
u
N
],
v
¼
[
v
1
,
v
2
,
v
3
,
...
,
v
N
], etc., including the zero n-tuple,
0
¼
[0, 0, 0,
...
, 0], and the
negative
n
tuple -
u
u
N
]. An
arithmetic n
-
space
consists of
the set
A
n
together with the additive and scalar multiplication operations defined by
u
¼
u
1
,
u
2
,
u
3
,
...
[
,
þ
v
¼
[
u
1
þ
v
1
,
u
2
þ
v
2
,
u
3
þ
v
3
,
...
u
N
þ
v
N
] and
a
u
¼
a
u
1
,
a
u
2
,
a
u
3
,
...
[
,
a
u
N
], respectively. The additive operation defined by
u
þ
v
¼
[
u
1
þ
v
1
,
u
2
þ
v
2
,
u
3
þ
v
N
] is the
parallelogram law of addition
. The parallelogram law
of addition was first introduced and proved experimentally for forces. A vector is
defined as an element of a vector space, in our case a particular vector space called
the arithmetic
n
-space.
The scalar product of two vectors in
n
dimensions was defined earlier, (A.31).
This definition provided a formula for calculating the scalar product
u
v
3
,
...
u
N
þ
v
and the
j¼
p
j¼
p
magnitude of the vectors
u
and
v
,
j
u
u
u
and
j
v
v
v
. Thus one can
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