0
votes
2answers
54 views

explain this confusing algebraic identity?

Can anyone show, step-by-step, how the expression on the LHS can be turned into the expression on the RHS? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
0
votes
2answers
30 views

Going from Linear algebra to Multivariable Calculus [closed]

I just finished a course in Linear algebra, can anyone tell me how Linear and multivariable calc are related?
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 ...
0
votes
1answer
30 views

Derivative of a path.

Define path: $s(t)= \langle 0,\cos(t),\sin(t)\rangle$ We are given that it is on the surface of $F(x,y,z)= x^2 + y^2 + z^2 -1$ Am told to find $s'(t)$ and a given point $t = (\pi/2)$. Is $s'(t) = ...
0
votes
1answer
16 views

Where do these Paths intersect

We have two paths: $r(t)=\langle cos(t),0,sin(t)\rangle $ $s(t)= \langle 0,cos(t),sin(t)\rangle$ where $t$ in $[0,pi]$ We are given that they are on the surface of $F(x,y,z)= x^2 + y^2 + z^2 -1$ ...
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 ...
0
votes
1answer
36 views

Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
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 ...
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 ...
0
votes
1answer
36 views

Hessian of a square root of a quadratic form

What is the Hessian matrix of the square root of a quadratic form: $\left(w^T H w\right)^{0.5}$? Got the gradient, $0.5 \left(w^T H w\right)^{-0.5} ( 2 H w)$, which gives numerically correct ...
1
vote
1answer
28 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
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 ...
2
votes
1answer
26 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
4
votes
1answer
131 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
0
votes
1answer
72 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
2
votes
1answer
31 views

Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x} $ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x} $ For what values of $a$, $E1$ and $E2$ have ...
0
votes
0answers
27 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
54 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
27 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 ...
3
votes
1answer
34 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 × ...
1
vote
2answers
39 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
155 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
29 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
32 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
78 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
30 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
39 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
23 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
38 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
22 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
96 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
1answer
1k 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 $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
1
vote
0answers
47 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
36 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
48 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
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
46 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
51 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
94 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 ...
7
votes
1answer
302 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
67 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
108 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
131 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
70 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
82 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
24 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
132 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
80 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. ...