2
votes
1answer
38 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
0
votes
1answer
28 views

Derivative of a function of a matrix $f(B) = x^T(AB)^ky$

I have a function of the form $f(B) = x^T(AB)^ky$ where $x$ and $y$ are column matrices, $A$ and $B$ are square matrices, and $B$ is a diagonal matrix, and $k$ is an integer constant. I want to find ...
2
votes
2answers
62 views

Question about Implicit function theorem

I was asked a simple question, show that $y+\sin y=x$ sets in the neighborhood of $(0,0)$ $y$ as a function of $x$, and find $\dfrac{dy}{dx}(0,0)$ Firstly, my naive solution would be: Since $lim_{y ...
1
vote
1answer
74 views

Determining the values of k for which the Matrix A has an inverse

I've been given this question in class, with the 3x3 matrix: 2 1 0 1 2 1 0 -3 k My job here is to find the values of k for which this matrix has an ...
0
votes
0answers
29 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
0
votes
1answer
62 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 ...
0
votes
2answers
47 views

Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
2
votes
1answer
70 views

Differentiate vector norm by matrix

I've been trying to perform the following differentiation of a neural network: $$\frac{\delta||h(XW)\alpha-y||^2}{\delta W} = \frac{\delta}{\delta W}\sum_i(h(XW)_i\alpha-y_i)^2$$ Where $X$ and $W$ ...
1
vote
1answer
32 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} ...
1
vote
2answers
48 views

What is the derivative of a vector with respect to its transpose?

I've already looked at Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$, but I wasn't able to find the direct answer to my question in that question. What is the value of $$\frac{d}{dx} ...
2
votes
1answer
37 views

Local maximality implies global maximality?

Let $S$ be the unit sphere in $\mathbb R^n.$ For a given $A\in\operatorname{M}_n(\mathbb R),$ define $f:\mathbb R^n\mapsto\mathbb R$ as $f(x)=\langle x,Ax\rangle.$ Suppose $a\in S$ is an element ...
5
votes
1answer
130 views

Extended matrix function

I have a continuous matrix-valued function $f:\mathbb{R}^d\mapsto {\cal M}_{k\times d}$, with $d<k$, such that $f(x)$ is full rank for all $x\in\mathbb{R}^k$. Can I extend this function to be a ...
0
votes
1answer
26 views

Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector. Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?
0
votes
3answers
90 views

Why is this 'obviously' positive semi-definite?

Here is a snapshot from a book I am studying. I learned all about positive semi-definiteness, and in fact I know that this matrix they are showing is in fact PSD. What I do not know is how they ...
2
votes
0answers
290 views

Expressing the determinant of a sum of two matrices?

can $$det(A + B)$$ be expressed in terms of $$det(A), det(B), n$$ where $A,B$ are $n$ x $n$ matrices? # I made the edit to allow n to be factored in
2
votes
1answer
45 views

Second Order Derivative of a function $f:R^2\to R^2$

The Exercise: My Work: Part 1: $$ Df=\left( \begin{array}{ccc} D_1f_1 & D_2f_1\\ D_1f_2 & D_2f_2 \\\end{array} \right) $$ $$f_1(x,y)=\sin x+\sin y$$ $$f_2(x,y)=\cos x+\cos y$$ $$ ...
2
votes
0answers
36 views

Continuity of the orthogonal matrix-valued function

Given $d<k$. Suppose that $H:\mathbb{R}^k\rightarrow {\cal M}(\mathbb{R})_{d,k}$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. Now define a ...
2
votes
0answers
28 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
25 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.
2
votes
2answers
40 views

transforming a vector from cartesian to spherical and cylindrical co-ordinate system

I know the formula(which i don't know how to copy here but it was in matrix form) for transforming a vector from cartesian system to spherical or cylindrical coordinate system. But, I want to know its ...
0
votes
1answer
72 views

Finding the determinant of an $n\times n$ matrix… and the inverse

Finding the determinant of a $2\times 2$ matrix is easy and the inverse is even easier. Finding the determinant of a $3\times 3$ matrix and its inverse is a little more difficult but still doable. ...
1
vote
0answers
59 views

The inverse matrix |${\delta}$| does it have an application

The jacobian is the determinant of |${\delta}$| this means that |${\delta}$| is invertible. Does this inverse have any use in the real world? Maybe I am not clear in my question: Does the inverse of ...
0
votes
1answer
48 views

How to derive curl in spherical coordinates

This is one of those questions where I know I am making a dumb mistake someplace and I am trying to check where it is. $$ \begin{vmatrix} \frac{\bf\hat r}{r^2\sin\theta} & ...
5
votes
2answers
96 views

Derivative of the trace of matrix product $(X^TX)^p$

Let $X$ be a squared matrix, We know that $\frac {\partial tr(X^TX)}{\partial X}$ is $2X$ But how about the case of $\frac {\partial tr((X^TX)^2)}{\partial X}$ or even $\frac {\partial ...
0
votes
1answer
80 views

Divergence calculation for jacobians

Suppose that u is suitably regular (e.g. $C^2(\mathbb{R}^N,\mathbb{R}^N)$ or $W^{1,2}(\mathbb{R}^N)^N$) and we write $$\det (\nabla u)=\nabla u^1 \cdot\sigma$$ for some $\sigma$ (obtained via the ...
0
votes
1answer
166 views

gradient of vector 2-norm

I have a function $f(\Theta) = \frac{1}{2N}\| y-\mathcal{X}(\Theta)\|_2^2$. Matrix $\Theta\in\mathbb{R}^{m_1\times m_2}$, $y=[y_1,\cdots,y_N]^T\in\mathbb{R}^N$ is the observation vector, and we use ...
0
votes
1answer
49 views

Conditions for linear independence of extended vector systems

Assume $$g: R^n \times R^m \rightarrow R^n$$ $$h: R^n \times R^m \rightarrow R$$ $$(x,y) \in R^n \times R^m$$ I would like to show that the following vectors are linearly independent: ...
1
vote
4answers
57 views

Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the ...
0
votes
0answers
44 views

Calculating the Jacobian for sampled data

I am reading about the Jacobian matrix which I interpret as a generalized gradient. I would like to take my investigations a bit further, so I have constructed some sample data which I would like to ...
1
vote
1answer
75 views

Is there a general form for the derivative of a matrix to a power?

Let $S:Mat(2,2) \rightarrow Mat(2,2)$ be the squaring map $S(A)=A^2$ then $[DS(A)]B=AB+BA$. I was wondering if there was a general form for this solution ($S(A)=A^n$, then $[DS(A)]B =$...). I have ...
1
vote
1answer
298 views

Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
0
votes
2answers
41 views

Non-elementwise Matrix Derivatives

Let A,B,C,D,X be matrices. I'd like to perform a Gradient Descent minimization to the loss functin $$ tr[(AXBX^TC-D)^T(AXBX^TC-D)] $$ My question is, how to take the gradient efficiently w.r.t. $B$? ...
0
votes
1answer
112 views

Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
0
votes
1answer
49 views

Change of variable formula, hermitian matrices

Let \begin{align} (d\mathbf{H})= \bigwedge_{1\leq j\leq k\leq N} d h_{jj}^{(1)} \bigwedge_{1\leq j< k\leq N}d h_{jk}^{(2)} \ ... \bigwedge_{1\leq j< k\leq N}\ d h_{jk}^{(\beta)} \end{align} ...
0
votes
1answer
92 views

Differentiation of a unitary matrix

Let $\mathbf{U}$ be a unitary matrix ($\mathbf{UU}^\dagger=\mathbf{1}$). What does this implies for $d( \mathbf{ U U }^\dagger)$? Is it mathematically sound to say: \begin{equation} d\mathbf{U} ...
1
vote
1answer
55 views

Derivative of a function of vector parameter. Problem with notation.

I have an error function $$err=\frac{1}{N}\left[\textbf{y}^T\ln{\textbf{x}}+(\textbf{1}-\textbf{y})^T\ln{(\textbf{1}-\textbf{x})}\right]$$ I need to find the gradient $\bigtriangledown_x{}err$, such ...
3
votes
1answer
48 views

Projection and direct sum

I want to show that for every projection $A^2=A$ we have that there exists a subspace $U_1 \subset ker(A)$ and $U_2$ such that $A|_{U_2} = id$ such that $V = U_1 \oplus U_2$. Does anybody here have a ...
7
votes
4answers
176 views

General question about matrix calculus with specific example (with attempted answer)

I'm struggling to find the right way to approach matrix calculus problems generally. As an example of a problem that is bothering me, I would like to calculate the derivative of $||Ax||$ (Euclidean ...
0
votes
1answer
47 views

How to Differentiate this Matrix product

I am trying to solve the matrix equations for linear regression and it leads me to the following differentiation. I cannot find an explanation on how to do it on the Internet, only the result being ...
0
votes
2answers
93 views

Linear Algebra- Matrix derivative

I have a question related to finding derivatives of matrices. What is the derivative of $$(A.X)*(X^n)*(X.B)$$ with respect to $x_{11}$ ? . is the element wise product * is matrix product $x_{11}$ is ...
2
votes
5answers
471 views

How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior ...
1
vote
1answer
78 views

Study of Matrix Calculus

I need to study matrix calculus such as integration, differentiation, differentiation of functions of determinants and inverse matrices and then also other matrix based calculations such as ...
2
votes
0answers
124 views

Inner product of two complex vectors?

Given $A \in \mathbb R^{m \times n}$real matrix. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n \times 1}$, can someone help me find the relationship between the following two ...
2
votes
2answers
74 views

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I'm not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,...,x_{n-1},x_n)$ to point $(0,...,0,-1)$ so that all the other points are also ...
0
votes
0answers
84 views

partial derivatives

how do you compute the first and second partial of $\Psi$ w.r.t $\alpha$,$\beta$ and $\lambda$ given \begin{equation*}\Psi ...
0
votes
1answer
124 views

What is this notation relating to the Jacobian matrix operation?

In the below image, in the very bottom-most equation is the partial differential at the end of the equation being multiplied to every element in the inverse Jacobian matrix (and then beta_n added)? Or ...
1
vote
1answer
168 views

Derivative of norm-infinity of vector

So I know that $\frac{dX}{dX} = \mathbb{I}$ where $X \in \mathbb{R}^n$ and $\mathbb{I} \in \mathbb{R}^{n \times n}$ is the identity matrix. Now, what is the following derivative? ...
0
votes
0answers
57 views

Differential of tr(ABA'C)

I have checked the question Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$? But I don't understand the proof in its entirety thats why I posted this question. In the ...
1
vote
1answer
88 views

Determining $u=v \times w$ using the cross product

Let $v = (3,0,0)$ and $w=(0,1,-1).$ Determine $u = v \times w$ using the geometric properties of the cross product rather than the formula. What are the possible angles $x$ between two unit vectors ...
0
votes
2answers
225 views

Jacobian Matrix?

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be given by $f(x,y)=(x^4-y^4,xy)$. (i) Evaluate the Jacobian of $f$, and its Jacobian determinant. (ii) Show that $f$ is locally invertible at any ...