trace (矩阵的迹) 的性质
性质1
tr(Am×nBn×m)=tr(BA)
t
r
(
A
m
×
n
B
n
×
m
)
=
t
r
(
B
A
)
证明
tr(AB)
t
r
(
A
B
)
=∑i=1m(AB)ii
=
∑
i
=
1
m
(
A
B
)
i
i
=∑i=1m∑j=1naijbji
=
∑
i
=
1
m
∑
j
=
1
n
a
i
j
b
j
i
=∑j=1n∑i=1mbjiaij
=
∑
j
=
1
n
∑
i
=
1
m
b
j
i
a
i
j
=∑j=1n(BA)jj
=
∑
j
=
1
n
(
B
A
)
j
j
=tr(BA)
=
t
r
(
B
A
)
性质2
tr(Am×pBp×nCn×m)
t
r
(
A
m
×
p
B
p
×
n
C
n
×
m
)
=∑i=1m(ABC)ii
=
∑
i
=
1
m
(
A
B
C
)
i
i
=∑i=1mai,:(BC):,i
=
∑
i
=
1
m
a
i
,
:
(
B
C
)
:
,
i
=∑i=1mai,:Bc:,i
=
∑
i
=
1
m
a
i
,
:
B
c
:
,
i
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