EE 103, Winter ’09, Prof SEJ:
HW 3 Sol.
Page 1 / 12
Applied Numerical Computing
Instructor: Prof. S. E. Jacobsen
HW3 Solution
Students: Distributed HW solutions are a component of the course and should be
fully understood.
Problem 1)
We have included “arrows” to remind you that the components, especially the “
i
b
”, are
vectors.
We know that we can add, transpose, and multiply
block matrices
if the corresponding
blocks have the correct sizes (and if we pay attention to the order of multiplication as we
multiply the blocks).
We write A and B as follows:
1
1
2
2
...
,
and
n
m n
n k
n
b
b
A
a
a
a
B
b
×
×
⎡
⎤
⎢
⎥
⎢
⎥
⎡
⎤
=
=
⎣
⎦
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
G
G
G
G
G
#
G
,
where
i
a
G
are
1
m
×
column vectors and
j
b
G
are
1
k
×
row vectors. A and B have the
form of two block matrices now and we can treat each block as if it were a number, and
as long as the dimensions match, we can proceed to multiply and sum.
Normally we have:
[
]
1
2
1
2
1
1
n
n
n
n
y
y
x
x
x
x y
x y
y
⎡
⎤
⎢
⎥
⎢
⎥
=
+
+
⎢
⎥
⎢
⎥
⎣
⎦
"
"
#
so, in the case of block matrices A and B, we can write:
1
2
1
2
2
1
2
1
...
.
n
n
n
j
n
j
j
n
b
b
AB
a
a
a
a b
a b
a b
a b
b
=
⎡
⎤
⎢
⎥
⎢
⎥
⎡
⎤
⇒
=
=
+
+
+
=
⎣
⎦
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
∑
G
G
G
G
G
G
G
G
G
G
G
G
G
"
#
G
the dimensions of each pair of matrices (row and column vectors) work for
multiplications and summation.
Each
i
j
a b
G
G
produces an
m
k
×
matrix and the result of
the sum will also be
m
k
×
as we expect.
Alternatively, the results can be shown as follows:

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