转置
(
A
+
B
)
T
=
A
T
+
B
T
(A+B)^T=A^T+B^T
(A+B)T=AT+BT
(
A
B
)
T
=
B
T
A
T
(AB)^T = B^TA^T
(AB)T=BTAT
逆
(
A
T
)
−
1
=
(
A
−
1
)
T
(A^T)^{-1} = (A^{-1})^T
(AT)−1=(A−1)T
(
A
B
)
−
1
=
B
−
1
A
−
1
(AB)^{-1} = B^{-1}A^{-1}
(AB)−1=B−1A−1
迹
t
r
(
A
T
)
=
t
r
(
A
)
tr(A^T) = tr(A)
tr(AT)=tr(A)
t
r
(
A
+
B
)
=
t
r
(
A
)
+
t
r
(
B
)
tr(A+B) = tr(A)+tr(B)
tr(A+B)=tr(A)+tr(B)
t
r
(
A
B
)
=
t
r
(
B
A
)
tr(AB) = tr(BA)
tr(AB)=tr(BA)
t
r
(
A
B
C
)
=
t
r
(
B
C
A
)
=
t
r
(
C
A
B
)
tr(ABC) = tr(BCA) = tr(CAB)
tr(ABC)=tr(BCA)=tr(CAB)
行列式
d
e
t
(
A
)
也
记
作
∣
A
∣
det(A)也记作|A|
det(A)也记作∣A∣
d
e
t
(
A
)
=
∑
σ
∈
S
n
p
a
r
(
σ
)
A
1
σ
1
A
2
σ
2
…
A
n
σ
n
det(A) = \sum_{\sigma \in S_n}{par(\sigma)A_{1\sigma_1}A_{2\sigma_2}\dots A_{n\sigma_n}}
det(A)=σ∈Sn∑par(σ)A1σ1A2σ2…Anσn
其中
S
n
S_n
Sn 为所有
n
n
n 阶排列（permutation）的集合，
p
a
r
(
σ
)
par(\sigma)
par(σ) 的值为
(
−
1
)
t
(
σ
1
σ
2
…
σ
n
)
(-1)^{t(\sigma_1\sigma_2\dots\sigma_n)}
(−1)t(σ1σ2…σn)
其中
t
(
σ
1
σ
2
…
σ
n
)
t(\sigma_1\sigma_2\dots\sigma_n)
t(σ1σ2…σn)
是
σ
1
σ
2
…
σ
n
\sigma_1\sigma_2\dots\sigma_n
σ1σ2…σn 的逆序数。
性质
d
e
t
(
c
A
)
=
c
n
d
e
t
(
A
)
det(cA) = c^ndet(A)
det(cA)=cndet(A)
d
e
t
(
A
T
)
=
d
e
t
(
A
)
det(A^T) = det(A)
det(AT)=det(A)
d
e
t
(
A
B
)
=
d
e
t
(
A
)
d
e
t
(
B
)
det(AB) = det(A)det(B)
det(AB)=det(A)det(B)
d
e
t
(
A
−
1
)
=
d
e
t
(
A
)
−
1
det(A^{-1})=det(A)^{-1}
det(A−1)=det(A)−1
d
e
t
(
A
n
)
=
d
e
t
(
A
)
n
det(A^n) = det(A)^n
det(An)=det(A)n
Frobenius范数
矩阵
A
∈
R
m
×
n
A \in \mathbb{R}^{~ m \times n}
A∈R m×n 的
F
r
o
b
e
n
i
u
s
Frobenius
Frobenius 范数定义为
∥
A
∥
F
=
(
t
r
(
A
T
A
)
)
1
/
2
=
(
∑
i
=
1
m
∑
j
=
1
m
A
i
j
2
)
1
/
2
\| A \|_F = (tr(A^TA))^{1/2} = (\sum_{i=1}^{m}{\sum_{j=1}^{m}{A_{ij}^2}})^{1/2}
∥A∥F=(tr(ATA))1/2=(i=1∑mj=1∑mAij2)1/2
容易看出，矩阵的
F
r
o
b
e
n
i
u
s
Frobenius
Frobenius 范数就是将矩阵张成向量后的
L
2
L_2
L2 范数。
