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首页矩阵运算速查手册:实用公式与技巧集锦
矩阵Cookbook是一份实用的在线资源,由Kaare Brandt Petersen和Michael Syskind Pedersen共同编撰,发布于2012年11月。该文档旨在为那些需要快速查阅矩阵运算相关知识的人提供一个方便的参考手册。它汇总了诸如恒等式、近似公式、不等式以及矩阵间的各种关系,这些都是数学和工程领域中常见的工具。 该文档的核心内容是收集了众多来源中的矩阵理论知识,包括但不限于互联网上的简短笔记和书籍附录。读者可以在这里找到基础的矩阵运算性质,如矩阵加法、乘法的性质,行列式的计算方法,矩阵逆的存在条件,以及与特征值、特征向量相关的公式。此外,还包括了一些矩阵近似算法和误差分析,这对于数值线性代数和信号处理等领域至关重要。 值得注意的是,矩阵Cookbook并不是原创作品,而是对已有的研究成果进行了整合。尽管如此,由于涉及的信息量大,难免会出现错误、排版问题或遗漏,作者们鼓励读者在发现这类问题时通过邮箱cookbook@2302.dk提供修正建议,以保持内容的准确性和完整性。 作为一本持续更新的资源,矩阵Cookbook反映了最新的研究进展和版本,通过查看页面顶部的日期可以判断当前版本。对于希望扩展或深入理解特定主题的读者,该文档也欢迎提出新的内容建议,以便进一步丰富和优化矩阵运算的学习资源。 关键词“矩阵公式”表明这本指南特别关注的是具体操作和定理,而不是抽象的理论探讨。矩阵Cookbook是一份实用且方便查询的参考工具,无论是学习者、教师还是研究人员,都能从中受益匪浅。
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2.5 Derivatives of Traces 2 DERIVATIVES
∂
∂X
a
T
(X
n
)
T
X
n
b =
n−1
X
r=0
h
X
n−1−r
ab
T
(X
n
)
T
X
r
+(X
r
)
T
X
n
ab
T
(X
n−1−r
)
T
i
(92)
See B.1.3 for a proof.
Assume s and r are functions of x, i.e. s = s(x), r = r(x), and that A is a
constant, then
∂
∂x
s
T
Ar =
∂s
∂x
T
Ar +
∂r
∂x
T
A
T
s (93)
∂
∂x
(Ax)
T
(Ax)
(Bx)
T
(Bx)
=
∂
∂x
x
T
A
T
Ax
x
T
B
T
Bx
(94)
= 2
A
T
Ax
x
T
BBx
− 2
x
T
A
T
AxB
T
Bx
(x
T
B
T
Bx)
2
(95)
2.4.4 Gradient and Hessian
Using the above we have for the gradient and the Hessian
f = x
T
Ax + b
T
x (96)
∇
x
f =
∂f
∂x
= (A + A
T
)x + b (97)
∂
2
f
∂x∂x
T
= A + A
T
(98)
2.5 Derivatives of Traces
Assume F (X) to be a differentiable function of each of the elements of X. It
then holds that
∂Tr(F (X))
∂X
= f(X)
T
where f (·) is the scalar derivative of F (·).
2.5.1 First Order
∂
∂X
Tr(X) = I (99)
∂
∂X
Tr(XA) = A
T
(100)
∂
∂X
Tr(AXB) = A
T
B
T
(101)
∂
∂X
Tr(AX
T
B) = BA (102)
∂
∂X
Tr(X
T
A) = A (103)
∂
∂X
Tr(AX
T
) = A (104)
∂
∂X
Tr(A ⊗ X) = Tr(A)I (105)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 12
2.5 Derivatives of Traces 2 DERIVATIVES
2.5.2 Second Order
∂
∂X
Tr(X
2
) = 2X
T
(106)
∂
∂X
Tr(X
2
B) = (XB + BX)
T
(107)
∂
∂X
Tr(X
T
BX) = BX + B
T
X (108)
∂
∂X
Tr(BXX
T
) = BX + B
T
X (109)
∂
∂X
Tr(XX
T
B) = BX + B
T
X (110)
∂
∂X
Tr(XBX
T
) = XB
T
+ XB (111)
∂
∂X
Tr(BX
T
X) = XB
T
+ XB (112)
∂
∂X
Tr(X
T
XB) = XB
T
+ XB (113)
∂
∂X
Tr(AXBX) = A
T
X
T
B
T
+ B
T
X
T
A
T
(114)
∂
∂X
Tr(X
T
X) =
∂
∂X
Tr(XX
T
) = 2X (115)
∂
∂X
Tr(B
T
X
T
CXB) = C
T
XBB
T
+ CXBB
T
(116)
∂
∂X
Tr
X
T
BXC
= BXC + B
T
XC
T
(117)
∂
∂X
Tr(AXBX
T
C) = A
T
C
T
XB
T
+ CAXB (118)
∂
∂X
Tr
h
(AXB + C)(AXB + C)
T
i
= 2A
T
(AXB + C)B
T
(119)
∂
∂X
Tr(X ⊗ X) =
∂
∂X
Tr(X)Tr(X) = 2Tr(X)I(120)
See [7].
2.5.3 Higher Order
∂
∂X
Tr(X
k
) = k(X
k−1
)
T
(121)
∂
∂X
Tr(AX
k
) =
k−1
X
r=0
(X
r
AX
k−r−1
)
T
(122)
∂
∂X
Tr
B
T
X
T
CXX
T
CXB
= CXX
T
CXBB
T
+C
T
XBB
T
X
T
C
T
X
+CXBB
T
X
T
CX
+C
T
XX
T
C
T
XBB
T
(123)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 13
2.6 Derivatives of vector norms 2 DERIVATIVES
2.5.4 Other
∂
∂X
Tr(AX
−1
B) = −(X
−1
BAX
−1
)
T
= −X
−T
A
T
B
T
X
−T
(124)
Assume B and C to be symmetric, then
∂
∂X
Tr
h
(X
T
CX)
−1
A
i
= −(CX(X
T
CX)
−1
)(A + A
T
)(X
T
CX)
−1
(125)
∂
∂X
Tr
h
(X
T
CX)
−1
(X
T
BX)
i
= −2CX(X
T
CX)
−1
X
T
BX(X
T
CX)
−1
+2BX(X
T
CX)
−1
(126)
∂
∂X
Tr
h
(A + X
T
CX)
−1
(X
T
BX)
i
= −2CX(A + X
T
CX)
−1
X
T
BX(A + X
T
CX)
−1
+2BX(A + X
T
CX)
−1
(127)
See [7].
∂Tr(sin(X))
∂X
= cos(X)
T
(128)
2.6 Derivatives of vector norms
2.6.1 Two-norm
∂
∂x
||x − a||
2
=
x − a
||x − a||
2
(129)
∂
∂x
x − a
kx − ak
2
=
I
kx − ak
2
−
(x − a)(x −a)
T
kx − ak
3
2
(130)
∂||x||
2
2
∂x
=
∂||x
T
x||
2
∂x
= 2x (131)
2.7 Derivatives of matrix norms
For more on matrix norms, see Sec. 10.4.
2.7.1 Frobenius norm
∂
∂X
||X||
2
F
=
∂
∂X
Tr(XX
H
) = 2X (132)
See (248). Note that this is also a special case of the result in equation 119.
2.8 Derivatives of Structured Matrices
Assume that the matrix A has some structure, i.e. symmetric, toeplitz, etc.
In that case the derivatives of the previous section does not apply in general.
Instead, consider the following general rule for differentiating a scalar function
f(A)
df
dA
ij
=
X
kl
∂f
∂A
kl
∂A
kl
∂A
ij
= Tr
"
∂f
∂A
T
∂A
∂A
ij
#
(133)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 14
2.8 Derivatives of Structured Matrices 2 DERIVATIVES
The matrix differentiated with respect to itself is in this document referred to
as the structure matrix of A and is defined simply by
∂A
∂A
ij
= S
ij
(134)
If A has no special structure we have simply S
ij
= J
ij
, that is, the structure
matrix is simply the single-entry matrix. Many structures have a representation
in singleentry matrices, see Sec. 9.7.6 for more examples of structure matrices.
2.8.1 The Chain Rule
Sometimes the objective is to find the derivative of a matrix which is a function
of another matrix. Let U = f (X), the goal is to find the derivative of the
function g(U) with respect to X:
∂g(U)
∂X
=
∂g(f(X))
∂X
(135)
Then the Chain Rule can then be written the following way:
∂g(U)
∂X
=
∂g(U)
∂x
ij
=
M
X
k=1
N
X
l=1
∂g(U)
∂u
kl
∂u
kl
∂x
ij
(136)
Using matrix notation, this can be written as:
∂g(U)
∂X
ij
= Tr
h
(
∂g(U)
∂U
)
T
∂U
∂X
ij
i
. (137)
2.8.2 Symmetric
If A is symmetric, then S
ij
= J
ij
+ J
ji
− J
ij
J
ij
and therefore
df
dA
=
∂f
∂A
+
∂f
∂A
T
− diag
∂f
∂A
(138)
That is, e.g., ([5]):
∂Tr(AX)
∂X
= A + A
T
− (A ◦ I), see (142) (139)
∂ det(X)
∂X
= det(X)(2X
−1
− (X
−1
◦ I)) (140)
∂ ln det(X)
∂X
= 2X
−1
− (X
−1
◦ I) (141)
2.8.3 Diagonal
If X is diagonal, then ([19]):
∂Tr(AX)
∂X
= A ◦ I (142)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 15
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