Tagged Questions

0
votes
0answers
24 views

$L:\mathbb{R}^2\to\mathbb{R}^2$ given by $L(x,y)=(x,-y)$ which of the following is true?

$L:\mathbb{R}^2\to\mathbb{R}^2$ given by $L(x,y)=(x,-y)$ which of the following is true? differentiable everywhere on $\mathbb{R}^2$ differentiable on $(0,0)$ only $DL(0,0)=L$ $ DL(x,y)=L$ for all ...
1
vote
1answer
41 views

calculate derivative for standard inner product

$L\colon\mathbb{R}^n\to \mathbb{R}$ $L_y(x)=\langle x,y\rangle$ for some inner product, $DL$ be the derivative of $L$. Its a Linear map so I know derivative will be itself only. I want to calculate ...
1
vote
0answers
19 views

Derivative methods for artifical neural networks with single hidden layer

I am trying to optimize the output of a given neural network with a single hidden layer. To accomplish this, I intend to find solve for all combinations of inputs where the derivative of the neural ...
3
votes
1answer
184 views

Derivative of a bra?

I understand that $$ \frac{\mathrm d}{\mathrm dt} \langle\psi|\psi\rangle =\left[\frac{\mathrm d}{\mathrm dt} \langle\psi|\right]|\psi\rangle + \langle\psi|\left[\frac{\mathrm d}{\mathrm ...
1
vote
2answers
41 views

Derivative of the linear functional given by an inner product form

$L\colon\mathbb{R}^n\to \mathbb{R}$ $L_y(x)=\langle x,y\rangle$ for some inner product, $DL$ be the derivative of $L$. Then which of the following is/are true? $DL(u)=DL(v)\quad \forall ...
0
votes
0answers
64 views

Hessian after coordinate changing

Let $f\colon \Bbb R^n\to\Bbb R$. Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$. Does anyone know a formula which express how the ...
3
votes
1answer
401 views

Matrix calculus : Find the gradient/derivative?

I know that the derivative of $Tr(Z^TAZ)$ w.r.t $Z$ is $2AZ$. Now I'd like to compute the derivative of $Tr\left[Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)Z\right]$ instead, w.r.t $Z ...
1
vote
1answer
65 views

Partial derivative with respect to a vector x for $F(x) = x^TA(x)x$

I have the next function $F(x) = x^TA(x)x$, where $x$ is a real vector with dimension $n$, and $A$ is a square real matrix $n \times n$ depending on the components of $x$. How can I compute the ...
1
vote
2answers
109 views

Derivative with respect to a matrix

How do we start with the matrix differentiation of this kind of equation? $$ V = [y_t-Cx_t]^TR^{-1}[y_t-Cx_t] $$ here $x_t$ and $y_t$ are vectors and $C$ and $R$ are matrices. R is a covariance matrix ...
1
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0answers
41 views

Neglecting solutions and reforming the system of differential equations with reducing the order but to keep choosen solutions

Here I have one problem which should help me to understand how to transform the system of differential equations with the condition to neglect two of four solutions and to get the appropriate system ...
2
votes
1answer
85 views

Proving a rule about del operator as applied to matrices

How can I prove the following easily?(If it is true of course.) \begin{align} \nabla_{\mathbf{x}_k} \left( \sum_{i=1}^{n}\sum_{j=1}^{n} \mathbf{x}^{T}_i \mathbf{W}_{ij} \mathbf{x}_j ...
7
votes
3answers
357 views

Derivative of determinant of a matrix

Good morning everyone, I would like to know how to calculate: $\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big)$ Help me please. Thank you
4
votes
3answers
107 views

Optimal symmetric rank-1 approximation

I want to find $\mathbf{x}$ that minimizes $\|A-\mathbf{x}\mathbf{x}'\|^2$ where $\|\cdot\|$ is Frobenius norm. Differentiating with respect to $\mathbf{x}$ and setting to $\mathbf{0}$, I get ...
2
votes
1answer
71 views

The derivative of characterestic polynomial?

Let $A\in M_{n}(R)$ and $f(x)$ be the characterestic polynomial of $A$. Is it true that $f'(x)=\sum_{i=1}^{^{n}}\sum_{j=1}^{n}\det(xI-A(i\mid j))$ which $A(i\mid j)$ is a submatrix of $A$ obtained by ...
3
votes
2answers
186 views

Derivative of a trace

I'm new here, so "Hi" to everyone :D I got the following problem. I have the matrices $A$, $B$, $C$, $X$ and $Y$. All matrices are square (say n-by-n). In particular: - $A$ is full rank - $B$ is ...
0
votes
1answer
52 views

What is the constraint in this LaGrange Multipliers ??

$x$ and $y$ are real numbers where satisfied the equation $x^2+y^2+xy-3x-3y-9=0$ Find the max. and min. values of $x^2+y^2$ I don't know how to find the constraint
0
votes
1answer
79 views

Lagrange method - non-linear system of equations

I have to compute optimal parametres of truncated cone so that its Volume is fixed (lets say it is 1) and its surface is minimal using Lagrange method These are equations desribing my object: ...
0
votes
1answer
86 views

Linear transformations in $P_n$

Consider the function $d/dx$ from $P_n$ (the real vector space of degree $\leq n$ polynomials in one variable $x$) to $P_{n-1}$. a) Prove that $d/dx$ is a linear transformation. b) Write the ...
1
vote
1answer
113 views

Derivative with respect to matrix

Let $F(m)=m^T m$ with $m$ a $n\times n$ matrix. I came across the statement $D_I F(m)=m^t+m$, where $D_iF$ means the derivative of $F$ at the identity matrix. I cannot understand how this emerges. I ...
0
votes
1answer
52 views

Proof for $df(X)/dX$ $f(X)=\operatorname{trace}{(AX^TB+C)^{-1}D}$

Denote $f(X)=\operatorname{trace}{(AX^TB+C)^{-1}D}$ and A,B,C,D are the constant matrix, X is the $R^{m*n}$ matrix. How to prove $df(X)/dX=-B(AX^TB+C)^{-1}D(AX^TB+C)^{-1}A$ I don't know the method ...
0
votes
1answer
103 views

Matrix Derivative

Can anyone please help me find the derivative of the ABC wrt B when: A is say 3*3 matrix B is 3*4 matrix C is 4*4 matrix. Thanks
1
vote
2answers
61 views

Finding direction of steepest descent on a hyperplane subject to orthogonal constraint

Given a linear objective function $f(\vec{x})=\sum_ia_ix_i$, the direction in which $f$ varies the greatest is known to be $\vec{\nabla}{f}$. Now given a non-zero vector $\vec{v}$, I am interested in ...
4
votes
1answer
85 views

Is this function convex when the input vector is positive?

I am wondering if $f(\mathbf{x})$ is convex on the input of a vector of $n$ positive reals $\mathbf{x}$: $$f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$$ where ...
1
vote
2answers
40 views

Where's the error in my calculation of a line through a point and being the tangent to a circle?

$$C:x^2+y^2=r^2$$ $$A(0,A_y)$$ I'd like to find the line L through A and being a tangent on C. Define point P on C. $$P(P_x,P_y)$$ $$P_x^2+P_y^2=r^2$$ Get the slope of L, by calculating the ...
5
votes
1answer
294 views

Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?

Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is: $$ \nabla_A \operatorname{trace}( ABA^{T}C ) = CAB + C^T AB^T $$ ...
1
vote
1answer
78 views

Vector derivatives, what is the minimum of this matrix equation?

I am new to vector derivatives and trying to figure out a lot for my Machine Learning course. I have given the following: $x \in \mathbb{R}^n$, $y \in \mathbb{R}^d$, $A \in \mathbb{R}^{d \times n}$, ...
2
votes
2answers
192 views

find the minimum value of $a+b+c$

There are natural numbers: $a$, $b$, $c$. $$\begin{cases} ab+bc+ca+\frac32(a+b+c)=5015,\\ 2abc-a-b-c=6366 \end{cases} $$ I need to find the minimum value of $a+b+c$. To my mind there's ...