1
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2answers
41 views

Derivative of the trace of $X^TP^TPX$ with respect to P

$\newcommand{\Tr}{\operatorname{Tr}}$ Consider the following expression: $\Tr(X^TP^TPX)$ where $X$ and $P$ are real matrices. What is the best way to approach the calculation of its derivative ...
1
vote
2answers
85 views

Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?

$A$ is a square matrix. All elements of $A$ depend on a parameter $t$, that is, $a_{ij}=a_{ij}(t)$. Let $S(A):=A^TA$, and take the derivative of $S$ w.r.t. $t$: $\displaystyle \frac{dS}{dt}$ Now, ...
1
vote
2answers
23 views

Matricial differentiation $x x^{\top} b $

What is the drivative of $x x^{\top} b $ with respect to x, knowing that b is constant vector?
0
votes
2answers
70 views

Matrix exponential Differentiation

We have the equation $e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$, where X is a matrix of dimension $3 \times 3$ . Now I have a function $f(x)=C_1x+C_2*\frac{x^2}{2} $ where $C_1,C_2,f(x)$ has ...
0
votes
2answers
28 views

Help me understand this different dimention matrix operation

I have $$ J(\theta) = \frac 1 {2m} (X \theta - \mathbf{y})^{\intercal} (X \theta - \mathbf{y}) $$ in which, $X$ is $m \times n$ matrix, $\theta$ is $n \times 1$ vector, and $\mathbf{y}$ is $m \times ...
0
votes
1answer
24 views

Differentiation involving determinant

This question has arisen by following the proof in the appendix of Louis Liporace's paper on maximum-likelihood estimation, where the paper concerns classes of probabilistic functions (elliptically ...
2
votes
1answer
24 views

Acquiring $Df(\mathbf{x})$

Sorry for the probably easy and silly question, but I try to teach myself linear algebra and I am stucked at "the derivative as a matrix" part. I know how to differentiate partially and I know how ...
0
votes
1answer
19 views

Complex function and Jacobian matrix

Given some complex-differentiable function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined $f(x,y)=u(x,y)+iv(x,y)$, we know the Cauchy-Riemann equations hold, so: $$\dfrac{\partial u}{\partial ...
3
votes
0answers
17 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
0
votes
1answer
28 views

Derivative of a Matrix with respect to a vector

I know that for two k-vectors, say $A$ and $B$, $\partial A/\partial B$ would be a square $k \times k$ matrix whose $(i,j)$-th element would be $\partial A_i/\partial B_j$. But could someone please ...
2
votes
1answer
42 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
1
vote
0answers
41 views

derivative of a matrix inverse

I wonder how to differentiate with respect to the diagonal matrix $X_d$, the following matrix : $$ X_d^T (\Sigma_d + X_d C X_d)^{-1} X_d $$ where $X_d$ and $\Sigma_d$ are diagonal matrices with ...
0
votes
1answer
36 views

Differentiation Matrix for central-difference scheme?

Central-difference scheme is defined to be: $f'(x) = \frac{f(x+d(x)) - f(x-d(x)))} {2*d(x)} + O(d(x)^2)$ Assume periodic boundary conditions, so that: $f(n+1)=f(1)$ I understand how to find all the ...
0
votes
1answer
32 views

What is the derivative of a skew symmetric matrix?

I'm trying to work out some Jacobians and I ran across a problem. If I have a function of a vector making it a skew symmetric matrix, like below, what is the derivative $f'$? $$ ...
1
vote
2answers
35 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
0
votes
1answer
30 views

Hessian matrix as derivative of gradient

From a text: For a real-valued differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix $D^2f(x)$ is the derivative matrix of the vector-valued gradient function $\nabla ...
1
vote
2answers
45 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
0
votes
1answer
36 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
0
votes
1answer
35 views

Element-wise derivative of the inverse of a matrix

I would appreciate if you could help me to obtain the element-wise derivative of $Z = (-A-BX)^{(-1)}$ where all of elements of $A$, $B$ and $X$ are positive. I conjecture that if I increase any of ...
0
votes
1answer
16 views

derivative of 2 norm wrt matrix

I have a matrix A which is of size m,n, a vector B which of size ...
1
vote
1answer
30 views

find the gradient of trace of the matrix

Prove that $\nabla_A Tr(AA^T) = 2A$, where A is any square matrix I did simple derivative with product rule,but i don't know where i messed up, I started with $\frac{\partial}{\partial A} ...
0
votes
0answers
51 views

Derivative of quadratic form w.r.t. matrix (product)

I need to show that some quadratic from: 1' A C A 1 is increasing in matrix C , where 1 is a (Kx1) vector of ones, and A and C are both (KxK) positive definite. Can I reason like this: 1) ...
1
vote
1answer
36 views

The derivative of $x^TAx$ w.r.t $t$

Suppose $P = x^TAx$ How to find $\frac{dP}{dt}$? if $x' = Bx$ , where $B$ has the same dimension as $A$. How to find the final answer? my answer is: $$\frac{dP}{dt} = 2[(A+A^T)x]x' = ...
1
vote
1answer
35 views

Differentiation of $u^{T}Su$

I want to differentiate $u^{T}Su$ wrt $u$ where $u$ is $n$ x $1$ and $S$ is $n$ x $n$matrix . So I did the following . Since $u^{T}Su$ is a number , I wrote its expression ie $$ f = ...
2
votes
1answer
53 views

Derivative of a Matrix to a Power

Fix a positive interger $k$ and let $F: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ be the map on $n \times n$ matrices defined by $F(A)= A^k$. Show that $F$ is differentiable at ...
1
vote
1answer
31 views

$\frac{d(X'X)}{dX}=?$

Thanks a lot for reading my thread. I am wondering what is the derivative of $X'X$ with respect to $X$? Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$; It would be great if ...
0
votes
2answers
85 views

Prove Derivative is sum of determinants

Given $n^2$ functions $f_{ij}$, each differentiable on an interval (a,b), define $F(x) = det[f_{ij}(x)]$ for each $x$ in $(a,b)$. Prove that the derivative $F'(x)$ is the sum of the determinants, $$ ...
1
vote
1answer
26 views

Represent derivation as a standard matrix (Linear mapping)?

Given a matrix $a$ of coefficients $\left( \begin{array}{cc} a_0 \\ a_1 \\ .. \\a_n\end{array} \right)$representing $a_0 + a_1 x + a_2 x^2 + ... a_n x^n$, how can I find a standard matrix D such that ...
0
votes
1answer
65 views

Check my answer - Differential of $P(A)=\det(A^{-1}-A)$

We are asked to find the differential of $P: GL_n(\mathbb R) \to \mathbb R$, $P(A)=\det(A^{-1}-A)$ and show it is differentiable. If we define $f(A)=\det(A)$ and $g(A)=A^{-1}-A$ then it is clear ...
1
vote
1answer
36 views

differential (Jacobi Matrix) of $f(A)=A^2$ where $A$ is a matrix - check my answer

I just want a quick verification that what I did here is correct: let $f(A)=A^2$ where $A$ is a n by n matrix with real entries. then $$D_f(A)=\lim_{t \to 0} \frac{f(A+tA)-f(A)}{t} = \lim_{t \to 0} ...
0
votes
1answer
65 views

Differentiation of the transpose of a vector? [closed]

Suppose $s$ is a scalar, and $x$ is a vector, how would I calculate $$ \left(\frac \delta {\delta x} (x^T s)\right) $$Basically I couldn't find any reliable source letting me know how to ...
0
votes
3answers
75 views

Derivative of matrix and vector in $\mathbf {v^TMv}$

Suppose I have a ($n\times 1$) vector $\mathbf v$ and a ($n\times n$) matrix $\mathbf M$ and I want to compute the derivative w.r.t. some $x$. Both $\mathbf v$ and $\mathbf M$ depend on the scalar ...
-2
votes
2answers
65 views

Vector derivative $\frac{d(Ax)}{d(x)}$ [closed]

I just need to know that whether it is $A$ or $A^T$ . I need it for an homework . Please be quick in telling me . Thanks !
1
vote
1answer
78 views

Derivative of matrix inverse w.r.t. vector

I need to differentiate the inverse of the $K\times K$ symmetric matrix $A$ w.r.t some vector (that $A$ depends on). Is there a rule for this? In case I do the derivative w.r.t. to some scalar there's ...
1
vote
1answer
63 views

Getting stonewalled on computation of $2\times 2$ Hessian matrix

The question: Let $z \in R^N$, and let $f(z) = \log[1^T z] \in R$. I am told that the Hessian matrix of this function is the following: $$ H = \frac{1}{1^Tz}\Big[ 1^Tz \mathrm{diag}(z) - zz^T \Big] ...
0
votes
1answer
68 views

Differentiation of bilinear form w.r.t. matrix

I need to do a derivative of bilinear form: b'C a w.r.t to Kx1 vector t where "b" and "a" are Kx1 vectors and "C" is KxK matrix that depends on vector t (and a and b are independent of t). Does anyone ...
0
votes
1answer
39 views

Show there is no solution…

Show that there is no solution to $(\bf D_n − I)p = 0$ except $\bf p = 0$; where $\bf D_n$ is the matrix representing the (first) derivative for degree $n$ polynomials and $\bf p=[c_0; c_1; c_2]$ ...
0
votes
2answers
38 views

Derivative of sqrt( Vector * Matrix * Vector ) according to one coordonate of the vector

I have a $n$ element vector $V$ and a symmetric $n\times n$ matrix $M$ (all of real elements). I calculate a score as $({ V^TMV })^{1/2}$ Now is there a formula that would give me: the derivative ...
0
votes
2answers
60 views

How to get the derivatives with respect to complex matrices

How could I get the derivative of the second term with respect to $\bar{\Delta}_k$ in the equation (19)? This result is obtained in the paper Robust Downlink Beamforming With Partial Channel State ...
3
votes
1answer
52 views

Vector by Matrix derivitive

According to wikipedia, there is no widely accepted definition of a Vector by Matrix derivative. I have a need of such a notion. For matrix w, and vector h. $$\mathbf{y=w \;h} $$ $$ ...
0
votes
1answer
28 views

simple partial derivative of constant times matrix?

Is the partial derivative of $cX$ w.r.t the real matrix $X$, given by $c$ or by $cI$, where $I$ is the identity, and $c$ is a constant scalar? please give a simple reasoning.
4
votes
3answers
284 views

Matrix derivative $(Ax-b)^T(Ax-b)$

I am trying to find the minimum of $(Ax-b)^T(Ax-b)$ but I am not sure whether I am taking the derivative of this expression properly. What I did is the following: \begin{align*} \frac{\delta}{\delta ...
3
votes
2answers
129 views

Matrix derivatives

Can someone please help me with this problem? ?I've already searched for similar examples in some linear algebra textbooks, but I couldn't find any... Thanks a lot! where $x\in \Bbb R^{n\times 1}$
4
votes
1answer
35 views

Computing derivative of function between matrices

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map ...
1
vote
0answers
75 views

derivative of determinat

For a lower trinagular, invertible but asymmetric matrix $X$, how to calculate the following: $$ \frac{\partial |XX^T|^{-1/2}}{\partial X} $$ I was doing the derivation, but not sure whether it was ...
1
vote
1answer
70 views

derivative of product of 2 inverse matrices

I was trying to differentiate the equation below: $$ \frac{\partial a^T X^{-T}X^{-1}a} {\partial X} $$ where X is invertible but not symmetric and $X^{-T}$ means transpose of inverse of X. In the ...
1
vote
3answers
78 views

Differentiating a non-linear functional with respect to a vector

I have the functional: $$F=v^T\times A \times v$$ Where $A$ is a function of $v$. The non-linear system of equations necessary to find $v$ is obtained doing: $$\frac{\partial F}{\partial v}=0$$ ...
2
votes
1answer
80 views

Understanding how to take derivatives with matrices

Currently we are doing 2nd order differential equations (we already did systems of homogenous two first order equations) and now that we have non-homogenous 2nd order equations we are doing method of ...
0
votes
1answer
299 views

Derivative (or differential) of symmetric square root of a matrix

Let A be a square, symmetric, positive-definite matrix. Let S be its symmetric square root found by a singular value decomposition. Let vech() be the half-vectorization operator. Is there a ...
1
vote
1answer
81 views

Question about the matrix representation of the differentiation map on the subspace generated by $\{1, t, e^{t}, e^{2t}\}$

As mentioned in a previous post (I think), I've been trying to learn some linear algebra, and so I've begun to post little questions whose answers I'm sure are obvious to most here; this is just a way ...