Information Technology Reference
In-Depth Information
21.2.2.5 Symmetric Matrix
n
×
n
holds
A
T
∈
R
=
If a square matrix
A
A
, it is called a
symmetric matrix
;if
it holds
A
T
n
×
n
,itis
=−
A
, it is called an
anti-symmetric matrix
.Given
A
∈
R
A
T
is anti-symmetric.
Therefore, any square matrix can be written as the sum of a symmetric matrix and
an anti-symmetric matrix, i.e.,
A
T
not difficult to verify that
A
+
is symmetric and
A
−
2
A
+
A
T
+
2
A
−
A
T
.
1
1
A
=
21.2.2.6 Trace
n
×
n
is called the trace of the
The sum of diagonal elements in a square matrix
A
∈
R
matrix, denoted by tr
A
:
n
tr
A
=
A
ii
.
i
=
1
The properties of trace are listed as below:
n
×
n
,wehavetr
A
=
tr
A
T
.
•
Given
A
∈
R
n
×
n
•
Given
A
∈
R
and
α
∈
R
,wehavetr
(αA)
=
α
tr
A
.
n
×
n
,wehavetr
(A
•
Given
A, B
∈
R
+
=
+
B)
tr
A
tr
B
.
•
Given
A
and
B
such that
AB
is a square matrix, we have tr
(AB)
=
tr
(BA)
.
•
Given matrices
A
1
,A
2
,...,A
k
such that
A
1
A
2
···
A
k
is a square matrix, we have
tr
A
1
A
2
···
A
k
=
tr
A
2
A
3
···
A
k
A
1
=
tr
A
3
A
4
···
A
k
A
1
A
2
=
···
=
tr
A
k
A
1
···
A
k
−
1
.
21.2.2.7 Norm
n
A
norm
of a vector is a function
f
:
R
→
R
that respects the following four
conditions:
n
•
non-negativity:
f(x)
≥
0,
∀
x
∈
R
•
definiteness:
f(x)
=
0 if and only if
x
=
0
n
•
homogeneity:
f(αx)
=|
α
|
f(x)
,
∀
x
∈
R
and
α
∈
R
n
•
triangle inequality:
f(x)
+
f(y)
≥
f(x
+
y)
,
∀
x,y
∈
R
n
Some commonly-used norms for vector
x
∈
R
are listed as below:
1
=
i
=
1
|
•
L
1
norm:
x
x
i
|
(
i
=
1
x
i
)
2
•
L
2
norm (or Euclidean norm):
x
2
=
•
L
norm:
x
∞
=
max
i
|
x
i
|
∞
