0
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0answers
25 views

Integration with matrices

I have written two equations in matrix format as follows $m(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} ...
0
votes
1answer
22 views

Scalar derivative of quadratic form where matrix depends on variable

I have the expression $$K(p(t),q(t)) = p^T D(q) p$$ Where D(q) is an n x n symmetric matrix, q and p are vectors (n x 1) depending on scalar variable t. I need to take the derivative of K with ...
1
vote
2answers
60 views

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then ...
1
vote
2answers
39 views

Calculus on Matrices [closed]

I have a basic doubt regarding calculus involving matrices. Dimensions of each matrices are also indicated along matrix name Question If I have a matrix $\kappa(s)_{3\times 1}$ what is ...
0
votes
0answers
20 views

inner derivation

I believe that I am doing this correctly but it seems too simple The instructions say to Let $R$ be a ring and a $\in R$. Define $da:R\to R$ For every $x\in R, da(x)=xa-ax$ I need to check (1) ...
0
votes
0answers
18 views

help with this derivative involving transformation matrix

I am working on an image analysis problem where I have a cost function which has the following term: $$ U = f(M(p)) $$ Here $f$ is an image which is sampled at discrete locations $p$. $M$ is a rigid ...
0
votes
0answers
26 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
0
votes
0answers
41 views

Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1 $. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
1
vote
0answers
50 views

How can I calculate this matrix differentiation?

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
0
votes
0answers
22 views

Matrix Algebra - Linear dependency

We have a given equation $ \frac{\mathrm{d}R(t) }{\mathrm{d} t}=R(t) \{(1-t)U_0+t U_1\}\tag 1$, all variables except scalar variable 't' has dimension $3 \times 3$. Given data $R(t)$ is ...
2
votes
0answers
68 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
5
votes
1answer
95 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
1
vote
2answers
62 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
89 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
25 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
78 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
30 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
25 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
23 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
18 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
39 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
43 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
37 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
36 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
50 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
43 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
51 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
37 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
49 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
21 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
35 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
61 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
38 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
37 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
54 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
106 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
27 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
37 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
72 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
78 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
66 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
101 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
65 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
77 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
40 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
40 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 ...