(from MathFlow) 设 A=(a
ij
)
, 且定义
试证: (1) ?
A
tr(AB)=B
t

; (2) ?
A
tr(ABA
t
C)=CAB+C
t
AB
t

.
证明: (1)
?
A
tr(AB)




=?
?
?

?a
ij


∑
m,n
a
mn
b
nm
?
?

=?
?
∑
m,n
δ
mi
δ
nj
b
nm
?
?

=(b
ji
)
=B
t
.




(2)
?
A
tr(ABA
t
C)





=?
?
?

?a
ij


∑
m,n,p,q
a
mn
b
np
a
qp
c
qm
?
?

=?
?
∑
m,n,p,q
δ
mi
δ
nj
b
np
a
qp
c
qm
+∑
m,n,p,q
a
mn
b
np
δ
qi
δ
pj
c
qm
?
?

=?
?
∑
p,q
b
jp
a
qp
c
qi
+∑
m,n
a
mn
b
nj
c
im
?
?

=?
?
∑
p,q
c
qi
a
qp
b
jp
+∑
m,n
c
im
a
mn
b
nj
?
?

=C
t
AB
t
+CAB.




[再寄小读者之数学篇](2014-05-27 矩阵的迹与 Jacobian),布布扣,bubuko.com
[再寄小读者之数学篇](2014-05-27 矩阵的迹与 Jacobian)
原文:http://www.cnblogs.com/zhangzujin/p/3755951.html