0
votes
0answers
2 views

Divisibility of a tuple of formal power series in multiple variables by a minor of their Jacobian matrix

My guess is that there's a general fact I might not be recalling going on here, but if not, here's some context: I've been reading an account of Artin Approximation by Herwig Hauser and Guillaume ...
0
votes
0answers
17 views

Powers of traces, integrals over spheres and class functions

Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, equipped with a Hermitian inner product $\langle \,\cdot\,,\,\cdot\, \rangle$. Let also $A$ be an endomorphism of ...
2
votes
1answer
41 views

A Question about Non-Conservative Vector Fields

In my multivariable calculus class, we spent some time discussing the vector field that was the gradient of arctan(y/x). This field was shown to be non-conservative in closed regions which enclosed ...
0
votes
1answer
19 views

Elements of a Negative Semidefinite Matrix

Use the definition of a negative definite matrix to show that if A is negative semi-definite: $$A_{ii} ≤ 0 \ \forall i $$ I know the definition (in terms of quadratic form) and the equivalent rules ...
1
vote
1answer
19 views

Cross Product in Levi-Civita Notation - The elementary basis vector's missing?

http://www.unl.edu.ar/ceneha/uploads/Cartesian_tensors_Index_notation_&_summation_convention.pdf avers: $1.$ $(a×b).(c×d) = \epsilon_{i jk}a_jb_k \quad e_{ilm}c_ld_m$ $2. \nabla × ...
0
votes
2answers
28 views

Proving $\nabla_A tr(ABA^T C) = CAB + C^T A B^T$

The above equation appears without proof on page 9 (equation 3) of Andrew Ng's notes on Machine Learning I have tried various approaches to prove this to no avail. From the notes it seems that it ...
0
votes
0answers
19 views

When is a function of two variables positive?

There are two functions, $u(t)$ and $v(t)$, $u:[0,\infty]\to {R}$ , $t=\frac{1}{2}(y_{1}^{2}+y_{2}^{2})$, $v=\frac{-uu'}{2tu'-u}$. They should satisfy the next: $u>0$ and $u+2tv>0$. If we ...
4
votes
2answers
137 views

How do you rearrange equations with dot products in them?

How can I go about rearranging an equation similar to this... $$\left(\pmatrix{-3\\0\\1}+ t \pmatrix{1\\4\\7} \right) \cdot n - a = 0$$ The issue I'm having is manipulating dot products
0
votes
0answers
22 views

multiplying Gaussian distributions of different dimensions

The multiplication of multivariate Gaussian distributions defined over some parameter vector of a given dimension can be achieved by the following. Assuming that the Gaussian is parametrized by the ...
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 ...
2
votes
1answer
35 views

Volume of ellipsoid using Linear Algebra

Can someone tell me how to find the volume of an ellipsoid of dimension $\mathbb{R}^3$ by using linear algebra? I know the formula is $\frac{4}{3}\pi abc$. I am given the equation ...
0
votes
1answer
28 views

Weird result with Multivariate Normal Distribution

Take zero mean MVN on k dimensions: $$ p(x) = \frac 1 {\sqrt {( 2 \pi ) ^k |\Sigma|}}e^{-\frac {1} {2} x \Sigma^{-1} x}$$ we will surely agree that $p(x)\leq1$ for all $x$, including $x=0$ ...
0
votes
1answer
29 views

Proving that a function is linear using the directional directive

I want to show that if for $f: \mathbb{R}^2 \mapsto \mathbb{R}^2$ if we have $f\begin{pmatrix} a_1 + v_1 \\ a_2 + v_2 \\ \end{pmatrix}$ = $f\begin{pmatrix} a_1 \\ a_2 \\ \end{pmatrix} + [Df(a)] ...
1
vote
0answers
34 views

Getting “semi” orthogonal basis from a linear independent set

Let $K_i: \mathbb{R}\mapsto \mathbb{R}^k$ are continuous functions for all $i=1,\dots,k-d$ such that for every fixed $t\in\mathbb{R}$ we have ${\cal K}_t=\{K_1(t),\dots,K_{k-d}(t)\}$ be a linear ...
0
votes
0answers
13 views

Matrix representation of second order differential of real function

So I'm trying to understand second order differentials of real functions. I've used this simple example $f(x,y)=x^2+y^2$ Now, the first order differential has the following matrix representation ...
0
votes
1answer
33 views

Differentiability of linear least squares

Show that least-squares $\|y-X\beta\|^2$ is twice differentiable and has minimizer. I understand that the second derivative is $X'X$. Also it is a composition of linear function which is ...
1
vote
1answer
21 views

Is this some sort of directional derivative problem I have here?

Let $V = y^2U_{1} - xU_{3}$. Also, let $f = xy$ and $g = z^3$ Compute $V[f]$ and $V[g]$. Now $U_{1} = (1,0,0)$ and $U_{3} = (0,0,1)$ Now in my notes, $V_{p}[f] = \displaystyle\frac{d}{dt}(f(p + ...
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
302 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
1
vote
0answers
42 views

Directional derivatives, linear maps, and uniform convergence

The Exercise Let $f(x,y)=x$ if $|y|>x^2$ and $f(x,y)=0$ otherwise. Show that all the directional derivatives of $f$ exist at the origin but there does not exist a linear map $D$ such that ...
0
votes
1answer
34 views

Proofs using vector properties

Let $a,b$ and $c$ be vectors in $\mathbb{R}^3$. How do I show that $$\|a-b\| \le \|a-c\|+\|c-b\|$$ and $$\|a \times b\|^2=\|a\|^2\|b\|^2-(a\cdot b)^2$$ ?
0
votes
1answer
42 views

the space of exterior k-forms is infinite dimensional. why?

let Z be an n-dimensional smooth manifold with smooth (n−1)-dimensional boundary ∂Z, representing the space of spatial variables. Denote by $Ω^k$(Z), k = 0, 1, . . ., n, the space of exterior k-forms ...
2
votes
1answer
74 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
41 views

Partial Differentiation Question. Solving when there is many variables

In one of my computer science classes we were given a homework problem that deals with partial differentiation. I never learned this in my math classes and have been trying to teach myself this but ...
1
vote
1answer
43 views

Did I solve this question about a line intersecting a plane correctly?

I'm asked to find if there is any point of intersection, and if so, where it is between the line represented by the symmetric equation $\frac{x-3}{3}=\frac{y+1}{-2}=\frac{z-10}{4}$ and the plane ...
4
votes
2answers
93 views

Show that if a function $f : \mathbb{R}^n \to \mathbb{R}^m$ is differentiable with differentiable inverse then $m = n$

So far I have: $\boldsymbol{f^{-1}} \circ \boldsymbol{f}(\boldsymbol{a}) = \boldsymbol{a} \implies [\boldsymbol{D}(\boldsymbol{f^{-1}}(\boldsymbol{a}) \circ \boldsymbol{f}(\boldsymbol{a}))] = I_n ...
6
votes
1answer
157 views

Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
3
votes
1answer
48 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
0
votes
2answers
50 views

degree of homogeneity

I have the function $$f(x,y)=\frac{y^b}{x^a}+\frac{x^b}{y^a}\quad a,b\gt0$$ The questions I have to answer are For which a and b is the function homogenous? Determine the degree of homogeneity My ...
2
votes
1answer
95 views

Derivative of $(Ax - b)^T(Ax-b)$

I am trying to take the derivative of $(Ax - b)^T(Ax-b)$ and setting it to zero without expanding the multiplication, by only using matrix calculus. I knew the partial derivative of $x^Tx$ according ...
0
votes
2answers
65 views

Does a plane have to be spanned by two vectors that are perpendicular?

I'm beginning to learn some vector calculus, and I am slightly confused about the textbook's explanation of planes spanned by two vectors. They said for example that the xy plane is an example of the ...
0
votes
1answer
65 views

Integrating two equations that equal, what happens to the constant on one of the sides?

In class, we were talking about Newton's 3rd law and how to integrate. $\int(g)dt = \int(y''(t))dt \implies g(t) + C = y'(t)$ I am confused about why the right hand side of the equation doesn't get ...
2
votes
0answers
18 views

Behaviour of Hessian under coordinate change

According to this one source http://www.math.ethz.ch/~pinkri/Theses/2008-Bachelor-Andreas-Steiger.pdf, the Hessian of a function $F: \Bbb K^n \rightarrow \Bbb K$ should change under a coordinate ...
6
votes
2answers
90 views

What is the interpretation of the eigenvectors of the jacobian matrix?

I'm trying to think about the jacobian matrix as a abstract linear map. What is the interpretation of the eigenvalues and eigenvectors of the jacobian?
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. ...
0
votes
0answers
19 views

An integral on a tensorial space

I need to solve the following integral: $\int\limits_{N_0\in v} N_0 K^2N_0rr^TN_0dN_0$ $N_0$ is an nxn symmetrical matrix (a tensor computed after an outer product $nn^T$), K^2 is an nxn matrix ...
0
votes
0answers
26 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 ...
0
votes
1answer
25 views

Get the closes cordinates to (0,0)

I'm making a system where I'm drawing a cube and I need to get the correct cordinates. Im trying to calculate the width and ...
0
votes
1answer
149 views

How to find plane equation by line and plane that perpendicular to

Find an equation for the plane that is perpendicular to the plane 2x +2y=1 and passes through the line ...
0
votes
0answers
29 views

Multivariable Calculus Question using change of basis

I was given the following problem: Let, $$F=<z-y,x-z,y-x>$$ Calculate the line integral of F around a circle that is lying on the plane $x+y+z=3$ that is of radius 4 centered at (1,1,1). The ...
1
vote
1answer
66 views

Vector equation involving $\nabla\dot{}(\overrightarrow A\times\overrightarrow B)$

I have to solve the following equation: $\nabla\dot{}(\overrightarrow u\times\overrightarrow v)=\overrightarrow u\dot{}\overrightarrow v$ where $\overrightarrow u$ and$\overrightarrow v$ are defined ...
1
vote
1answer
58 views

Proving the image set is $E^n$

Function $\mathbf g$ is $C^{(1)}$, and $\exists$ ${c\gt0}$ $\forall$ $s,t\in \mathbb R^n$ such that $|\mathbf{g(s)}-\mathbf{g}( t)|\geq c|\mathbf{s}-\mathbf{t}|$. It can be proved that it it is ...
0
votes
1answer
76 views

Is there a relationship between Matrix norm and the Jacobian of same matrix?

The exercise asks us to show that a function $\mathbf{g}$ that is $C^{(1)}$, and $\displaystyle \exists_{c\gt0} \forall_{\mathbf{s,t}\in \mathbf{R}^n} |\mathbf{g}\left( ...
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: ...
15
votes
3answers
721 views

Maximizing a sum of inner products

Someone asked this question on a French maths forum here and it caught my attention. The question is the following: let $(E, \langle \cdot, \cdot \rangle)$ be a Euclidean vector space. Find the ...
1
vote
0answers
93 views

Partial Derivative of a Scalar-product Resulting from Vector Multiplication

I am trying to differentiate the function below, but I am running into problems due to the point-wise multiplication with the matrix. $f(x,y,A) = (x^{T}y) \cdot A$ Where $\cdot$ denotes point-wise ...
0
votes
1answer
47 views

If $T$ is injective then there exists $\alpha>0$ such that $||Tx||\geq \alpha||x||$

Is this proof correct? I'm proving that if $T$ is a linear operator whose is injective then exist $\alpha>0$ such that $$||Tx||\geq\alpha||x||$$ for all $x$. By contrapositive. Assume that for all ...
2
votes
1answer
29 views

Question about proof of Morse Lemma

I am working on a problem for my differential geometry course. We are proving the following special case of the Morse lemma: Let $U \subseteq \mathbb{R}^n$ be open and containing the origin ...
2
votes
1answer
118 views

Geometric Interpretation of Jacobi identity for cross product

Is there a geometric "reason" for the Jacobi identity for cross products? Some geometric equality of some area ...? All proofs I know work by some form of linear algebra (or use the interpretation as ...
1
vote
0answers
36 views

How do you know what a 3-variable equation represents geometrically or physically?

Example equation: $$z^2+xy-2x-y^2=1$$ I know that equation represents a plane. Is there any easy way to tell what a 3-variable equation represents from just looking at it?