Graphics Reference
In-Depth Information
Figure C.2.
The triangle inequality.
q
|qr|
|pq|
r
|pr|
p
2
uv
+=<+ +>
uvuv
,
2
2
=+<
u
2
u v
,
+
v
2
2
£+
uuv
2
+
v
2
(
)
=+
uv
and equality holds only if
u
and
v
are linearly dependent.
The geometric content of the triangle inequality is that the sum of the lengths of
two sides of a triangle is larger than the length of the third side (see Figure C.2) and
is summarized in the next corollary.
If
p
,
q
,
r
Œ
R
n
, then
C.2.3. Corollary.
pr
<
pq
+
qr
unless
p
and
q
and
r
are collinear.
C.3
Matrices of Linear Transformations
We begin with a brief summary of basic facts dealing with matrices. See, for example,
[John67], [Lips68], or [NobD77] for more details of proofs.
Definition.
An n ¥ n matrix (a
ij
) over the reals is said to be
symmetric
if a
ji
= a
ij
for
all i and j. An n ¥ n matrix (a
ij
) over the complex numbers is said to be
Hermitian
if
a
ji
= a
ij
for all i and j. An arbitrary n ¥ n matrix (a
ij
) is said to be
diagonal
if a
ij
= 0 for all
i π j.
