2
votes
2answers
40 views

About matrix derivative

Suppose $A$ is a matrix with order n*n. we have the following equity but I don't know why. $f(x)=\frac{1}{2}x^TAx-b^Tx$. then $f'(x)=\frac{1}{2}A^Tx+\frac{1}{2}Ax-b$ Is there any rule like scalar ...
0
votes
0answers
19 views

Differentiation of cost function in adaptive CFO estimator

I'me trying to simulate the steepest descent algorithm for CFO estimation using null subcarriers (OFDM wireless). And some mathematic difficulties have arised. In the core of algorithm lies cost ...
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 ...
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 ...
2
votes
0answers
18 views

Rank of the differential

Let $f:\mathbb R^n \to \mathbb R^n$ such that $f$ maps roots of a polynomial to its coefficients. Meaning: if $(x-x_1)(x-x_2)...(x-x_n)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_n$ then $f\begin{pmatrix} x_1 ...
0
votes
1answer
24 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
41 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
40 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 ...
1
vote
1answer
37 views

Revisiting the product rule for derivatives

Let $E=C^{\infty}(\mathbb R, \mathbb R)$ Consider a linear transformation on $E$: $\delta$ such that $\forall f, g \in E, \delta(fg) =g\delta(f) +f\delta(g)$ Prove that there is some ...
2
votes
1answer
50 views

Derivative of determinant, which is correct?

I've seen two different results on the derivatives of determinants of matrices: $$\frac{\partial |X|}{\partial X_{ij}}=X_{ij}.\tag1$$ $$\frac{\partial\det(X)}{\partial X}=|X|(X^{-1})^{T}.\tag2$$ ...
0
votes
2answers
35 views

Re-writing a a differential function

I don't understand the concept of this... how do I derive a an equation written in terms of a function? How do I differentiate f(function inside) ...?
0
votes
0answers
33 views

Finding the constant of a function in terms of the gradient of a tangent.

Let $f : \Bbb R \to \Bbb R, f (x) = e^x+ k$, where $k$ is a real number. The tangent to the graph of $f$ at the point where $x = a$ passes through the point $(0, 0)$. Find the value of $k$ in terms of ...
1
vote
2answers
38 views

Constructing a matrix that computes derivatives

Consider the subset of functions given by $S = \text{Span}(e^{2t}\, \sin\, 3t, e^{2t}\, \cos\, 3t)$: Show that the derivatives of $e^{2t}\, \sin\, 3t$ and $e^{2t}\, \cos\, 3t$ are also in $S$ and ...
0
votes
0answers
21 views

Limits of norms and deriviative as linear transformation

I'm self-studying Spivak's Calculus on Manifolds and he introduces the derivative by first looking at it as a linear transformation, $Df(a) = \lambda$, saying that for a differentiable function ...
0
votes
1answer
34 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 ...
1
vote
1answer
29 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
24 views

5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
0
votes
0answers
45 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) ...
8
votes
4answers
870 views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
1
vote
1answer
66 views

Can Moore–Penrose pseudoinverse solve for underdetermined linear system?

Thanks for reading my thread. I am thinking, many of us know that Moore–Penrose pseudoinverse can solve for overdetermined system $Ax=b$, where $x=(A^TA)^{-1}A^Tb$; for exmplae the linear regression ...
1
vote
2answers
39 views

Linear operator exists then differentiable?

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such ...
0
votes
1answer
30 views

Linear transformation from $R^2$ to $R^2$.

Let $\vec{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\vec{f} (\vec{x}) = (x+y^2, x^3+5y)$ and $\vec{x} = (x,y) \in \mathbb{R}^2$. Let $\vec{h} = (h_1, h_2)$ and $\vec{a} = (1,1) \in ...
0
votes
1answer
69 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
0
votes
0answers
41 views

Calculus and Matrices

Suppose I have a linear operator $T: \mathbb{R} \rightarrow \mathbb{R}$, and also suppose that it's a composition of elementary functions, so its derivative, $T'$, is reasonable easy to find. I can ...
1
vote
0answers
60 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by producing a linear function

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
0
votes
2answers
80 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, $$ ...
0
votes
2answers
35 views

Linear Approximations

Can't figure out where I'm going wrong here. Isn't it just f(x)+f`(x) dx?
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 ...
1
vote
0answers
34 views

“Painless Conjugate Gradient”: alpha minimizes f when the directional derivative $\frac{df(x1)}{d\alpha} = 0$

I am reading the "Painless Conjugate Gradient Method" http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf On page 6, after equation 9, the author states "From basic calculus, $\alpha$ ...
0
votes
2answers
36 views

Maximum and Minimum Values of the function

What will be the maximum and minimum value of the following function, $f(x,y)=3x+4y$ in the region $0\le x \le1$, $-1\le y \le1 $
0
votes
1answer
20 views

Defining the function near some points.

How we can show that , $$\ln(x+2y)+32x^3y^2=\frac{1}{4}$$ defines $y$ as a function of $x$ near the points $\displaystyle\left(\frac{1}{2},\frac{1}{4}\right)$ and calculate $y'(1/2).$
2
votes
2answers
41 views

How is the derivative with respect to vector is taken in linear regression?

In the book I am studying the author motivates that the sum of the distances of data points to the fitted line can be written in matrix form as $$ (t-X\beta)^T(t-X\beta) $$ where X is a matrix that ...
1
vote
3answers
32 views

derivative of quadratic function without transposes

I'm trying to solve an equation of the following form: $$ \frac{\partial}{\partial X} A'XA'X $$ where $X$ and $A$ are both equal-length column vectors (and so that $A'XA'A$ is scalar). From looking ...
2
votes
1answer
60 views

Prove that $DT = I_v$, $TD \neq I_v$, where $D$ = differentiation operator and $T$ is integration

Let $V$ be the linear space of all real polys $p(x)$. Let $D$ denote the differentiation operator, and let $T$ the integration operator that maps each polynomial $p$ onto the polynomial $q$ given by ...
0
votes
1answer
64 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 ...
2
votes
2answers
151 views

Solve the following differential equation: $xy' - y = x^2$

I'm preparing to exam in Linear Algebra $2$ and I have problems with differential equations.. For example, the following exercise: Solve the following differential equation: $xy' - y = x^2$. I ...
1
vote
1answer
73 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
0answers
35 views

Quotient rule for the Jacobian

Is there an analog to the quotient rule that can be applied to the calculation of the Jacobian? Example: Can the jacobian of a quotient of two functions be decomposed into some series of linear ...
2
votes
1answer
86 views

Derivative of a Linear Map

I'm devastatingly incompetent at linear algebra and multivariable calculus. I just cannot understand it at all. Here's the easiest problem from my homework, and my attempt at solving it, and where I ...
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
0answers
16 views

derivative of $\nabla_{\lambda_i^{(1)}} \mathrm{Tr}\left((\Sigma_1 + \Sigma_2)^{-1} \cdot V^{(1)}_i V^{(1)^\mathrm{T}}_i \right) $

As stated in the title, I seek to find the derivative of Equation (1) with respect to the eigenvalues of the first covariance matrix, $\Sigma_1$. \begin{equation} \nabla_{\lambda_i^{(1)}}\; ...
1
vote
0answers
37 views

Books or readings on topics like differentiating a matrix

I recently encountered an increasing number of questions that requires operations like differentiating a matrix wrt a vector etc. I've never had any lecture on this and can't find any book relating to ...
0
votes
0answers
32 views

Partial derivatives in linearisation.

I'm working through a linearisation of the following system of equations \begin{align} \begin{split} u_t^{+}+\gamma u_x^{+}&=\mu(u^{+},u^{-})(u^{+}-u^{-}), \\ u_t^{-}+\gamma ...
1
vote
1answer
86 views

Proof that the derivative is unique?

Given a subset $\Omega$ of $\mathbb{R}^n$ and a function $\sigma: \Omega \to \mathbb{R}^n$, we define its derivative at $x$ to be a linear operator $\sigma'$ such that $$ \lim_{y \to 0} ...
5
votes
2answers
97 views

Derivative of a map involving the matrix inverse

I have $f: U\rightarrow \mathbb{R}$, $f(X):=\operatorname{tr}(X^{-1})$, $U$ contains all matrices $X$, which are positive definite and symmetric. I want to show that $f$ is differentiable on $U$. To ...
1
vote
0answers
36 views

Working out minimum of matrix equation

Basic linear algebra question. I am following some notes which goes directly from $$q(\gamma)=\inf_x\left[{1\over2}X^TDX+c^TX+\gamma^T(A^TX-b)\right]$$ to ...
1
vote
1answer
87 views

Questions on “painless conjugate gradient”: take gradient of a quadratic form

I am reading this paper: http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf I have difficulties on the derivation of equation (6) on page 4. It is to take gradient of a quadratic ...
0
votes
1answer
277 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
80 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 ...
1
vote
0answers
86 views

How to show that the first derivative is bounded in a function

\begin{equation} y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber \end{equation} How to show in above function the ...