Biomedical Engineering Reference
In-Depth Information
Problems
A.3.1.
Simplify the following expression by using the Einstein summation index
convention for a range of three:
¼
r
1
w
1
þ
r
2
w
2
þ
r
3
w
3
;
0
C ¼ð
u
1
v
1
þ
u
2
v
2
þ
u
3
v
3
Þð
u
1
v
1
þ
u
2
v
2
þ
u
3
v
3
Þ;
A
11
x
1
þ
A
22
x
2
þ
A
33
x
3
þ
f ¼
A
12
x
1
x
2
þ
A
21
x
1
x
2
þ
A
13
x
1
x
3
þ
A
31
x
1
x
3
þ
A
23
x
3
x
2
þ
A
32
x
3
x
2
:
A.3.2. The matrix
M
has the numbers 4, 5,
1 in its
second row and 7, 1, 1 in its third row. Find the transpose of
M
, the
symmetric part of
M
, and the skew-symmetric part of
M
.
5 in its first row,
1, 3,
Prove that
@
x
i
@
A.3.3.
x
j
¼ d
ij
.
A.3.4. Consider the hydrostatic component
H
, the deviatoric component
D
of the
symmetric part of
A
, and the skew-symmetric component
S
of the square
n
by
n
matrix
A
defined by (A.17) and (A.18),
and
S
tr
A
n
1
2
2
tr
A
n
1
2
½
A
T
A
T
H
¼
1
;
D
¼
A
þ
1
¼
A
:
Evaluate the following: tr
H
,tr
D
,tr
S
, tr(
H
D
)
¼
H
:
D
, tr(
H
S
)
¼
H
:
S
,
and tr(
S
D
)
¼
S
:
D
.
A.3.5.
For the matrices in example A3.3 show that tr
A
B
¼
tr
B
A
¼
666. In
general, will
A
:
B ¼ B
:
A
, or is this a special case?
A.3.6. Prove that
A
:
B
is zero if
A
is symmetric and
B
is skew-symmetric.
A.3.7. Calculate
A
T
B
T
, and
A
T
B
T
for the matrices
A
and
B
of Example
B
,
A
A.3.3.
A.3.8.
Find the derivative of the matrix
A
(
t
) with respect to
t
.
2
4
3
5
:
t
2
t
sin
o
t
A
ð
t
Þ¼
cosh
t
ln
t
17
t
t
2
ln
t
2
1
=
t
1
=
B
)
T
B
T
A
T
.
A.3.9.
Show that (
A
¼
C
)
T
C
T
B
T
A
T
.
A.3.10. Show that (
A
B
¼
A.4 The Algebra of n-Tuples
The algebra of column matrices is the same as the algebra of row matrices. The
column matrices need only be transposed to be equivalent to row matrices as
illustrated in equations (A.4) and (A.5). A phrase that describes both row and
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