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1answer
16 views

What is special about a transformation if the matrix of that transformation is symmetric?

If the matrix of a linear transformation T$\colon \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to ...
0
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0answers
12 views

Mapping values to logarithmic-like scale with adjustable “linearity factor”?

I have a stream of numbers in a, say, [0.1, 100] range. I need to display the number for a human (e. g. in a progress bar-like linear indicator), and I know that the distribution of the numbers is not ...
1
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0answers
13 views

Questions about a special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
0
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1answer
21 views

Finding the transformation matrix R

Please help me in solving this problem, I am not sure what a transformation matrix R is and how to proceed.. Any help is appreciated. Find the transformation matrix R that relates the (orthonormal ) ...
0
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1answer
28 views

finding if a linear transformation exists, and proving it.

We just started the topic of linear transformations and I have this hw question that I just don't understand. Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, ...
1
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0answers
22 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, ...
1
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0answers
12 views

A conformal mapping from a sector to a strip

What is the simplest function that maps the sector $r < 1$, $0 < \theta < \pi$ conformally onto the strip $0 < u < \pi/2$, $v > 0$? Here, $r$, $\theta$, $u$, $v$ have their usual ...
0
votes
1answer
19 views

Inverse Laplace transformation of (s^2-4s-2)/((s^2+2)^2)

I approached this problem as follow: $1.$ rewrote $(s^2-4s-2)$ into $(s-2)^2-6$ $2.$ Now break the function into 2 parts: $\frac{(s-2)^2}{(s^2+2)^2} + \frac{6}{(s^2+2)^2}$ the Laplace inverse ...
0
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3answers
22 views

“compression” transform

Is there a mathematical transform that cuts off a signal at two extreme values? Here is code to do what I want: ...
0
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0answers
22 views

Calculate rotation and translation of object from corresponding points. NOT affine transformation

I have measured 4 3D points X and corresponding 4 3D points Xp after rotation and translation of object. From equation ...
0
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0answers
32 views

Nullity and rank of a linear transformation

Let $T:V\to W$ be a linear transformation, where $$T=d^2/{dx}^2,\\V=\{f(x): f \text{ polynomial of degree}\leq n\},\\W=\{f(x): f \text{ polynomial of degree}\leq n-2\}.$$ What is the nullity of ...
1
vote
0answers
27 views

Legendre transform concave function

Let $f$ be a concave function and define $f^*(y) := \inf_{x}(yx-f(x))$. Is this in any sense related to the Legendre transformation? -If yes, is $f^*$ also concave? Is this transformation invertible ...
0
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0answers
9 views

Deriving the $F_3$ type generating function in Hamiltonian formulation

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...
0
votes
1answer
18 views

If S and T are transformattion mappings, what is [ST]?

S and T are transformation mappings, what does [ST] and [TS] mean? Does it mean transform via S and then apply T to the result and vice versa?
0
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0answers
28 views

Unimodularity of spin transformation

Consider a spin transformation \begin{equation} \zeta \to \tilde\zeta=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta} \end{equation} with all quantities being complex. It is said that without loss of ...
0
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1answer
26 views

Conformal mapping of nonsimply connected domains

The question asks: Map the complement of the arc $|z|=1$, $y\geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other. How would you construct such conformal ...
1
vote
1answer
79 views

Linear Algebra - - Linear transformation

The matrix $$ A=\left[\begin{array}{ccc} 1 & -a & a \\ -1 & a & a+2 \\ 1 & 2a+3 & -3a-4 \end{array}\right], $$ where $a \in \mathbb{R}$, represents a linear transformation $T: ...
0
votes
1answer
16 views

Let $X_1$ and $X_2$ be independent $n(0,1)$ random variables. Find the pdf of $(X_1-X_2)^2/2$.

I understand that $(X_1-X_2)/\sqrt2)$ ~ $n(0,1)$ since it is a linear combination of $X_1 $ and $X_2$ and hence $(X_1-X_2)^2/2$ ~ $\chi^2_1$. I'm having trouble on how to prove/show this ...
0
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1answer
28 views

Why does the discrete cosine transform as matrix multiplication work this way?

I have read that the DCT can be computed as a matrix multiplication. The 8x8 DCT matrix is: $D=\frac{1}{2}\left[\matrix{ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & ...
0
votes
0answers
16 views

Transformation of binomial distribution

If X ~ binomial(10, 1/3), find the pmf of Y, where Y = X^2. If my understanding is correct, pmf p_y(m) where Y is some function g(X) and m is number of successes desired is equal to p_x(g^-1(y)). ...
0
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0answers
100 views

Correctly transforming ODEs

Consider the following two systems of differential equations: \begin{eqnarray*} & k_{1}x'=k_{2}-\frac{[x(y+1)]^{2}}{k_{3}+[x(y+1)]^{2}}\\ & y'=k_{4}\frac{[x(y+1)]^{2}}{k_{3}+[x(y+1)]^{2}}-y ...
0
votes
2answers
32 views

Find invariant points, how to express using parameter

I have a matrix $$\begin{pmatrix}0&-1\\1&2\end{pmatrix}$$ where I have to find the invariant points for a transformation using this matrix. I have no problem working through to two ...
1
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2answers
24 views

inverse of similarity transformation

If $S$ is a similiarity transformation, i.e. there exists $c>0$, such that $$ \lvert S(x)-S(y)\rvert = c\lvert x-y\rvert. $$ Then, apparently, we have that $$ \big\lvert ...
2
votes
1answer
47 views

Odd polynominal

Let's define an odd polynominal be a polynominal which has odd degree, and ALL of its terms have odd exponential (except the constant), for example: $x^5+x^3+1$, or $x^7+2x^5+3x^3+4x+5$. We all know ...
0
votes
2answers
42 views

double integral over an arbitrary triangle

Assume we have an arbitrary triangle ABC in x-y plane and we want to integrate a function $f(x,y)$ over surface of this triangle as shown in fig. 1: We can define another coordination system [x' ...
1
vote
2answers
56 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
0
votes
0answers
17 views

Complex form of Fourier series - help

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ $$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$ ...
5
votes
1answer
67 views

How to transfrom my equation to $Y=KX^2$

In general , $$\vec{C}(u)=\vec{a_0}+\vec{a_1} u+\vec{a_2} u^2$$ is a parabolic arc between the points $\vec{a_0}$ and $\vec{a_0} + \vec{a_1} + \vec{a_2}$. So I'd like to prove it by myself: My trial ...
0
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0answers
22 views

General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
0
votes
1answer
23 views

What is the new point reflected with a respect to fixed line through origin?

If I have $(x,y)$, what would be the reflected point with respect to fixed line through origin? does it depend on what line I have, If I have $x$ or $2x$ $3x$, they all pass through origin.
0
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0answers
14 views

How to implement the Integer Wavelet Transform for images?

I have a description of a wavelet transform, but I am unsure on how to implement the algorithm based on the information given: $A_i,_j = ((I_{2i,2j} + I_{2i+1,2j}) / 2 )_{floor}$ $V_i,_j = ...
2
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1answer
23 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
0
votes
1answer
21 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
-1
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0answers
13 views

World- Camera-Projector coordinates using respective extrinsic matrices

When calibrating a camera using a library such as OpenCV, you get as a result the intrinsic matrix, distortion coefficients and numerous extrinsic matrices depending on the number of "chessboard ...
0
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0answers
13 views

How to calculate discrete cosine transform for a matrix

I have a 8x8 matrix and I want to calculate its discrete cosine transform (DCT-II). I have this formula but I don't know to use it with a matrix. In the French Wikipedia they gave an example for ...
3
votes
2answers
59 views

Prove that an element of the basis is an element of the Kernel after linear transformation

Let $T:R^4\rightarrow R^4$ and basis $B=(v_1,v_2,v_3,v_4)$. $$T(v_1)+T(v_2)=T(v_3)\; \text{ and } \; T(v_1)+T(v_3)=T(v_2)$$ Prove that $v_1\in Ker(T)$ What I wrote is: $$T(v_1)=T(v_3)-T(v_2)\; ...
0
votes
2answers
41 views

Transformation Matrix project

My task is to find the Transformation Matrix, that projects, any point of the xy-plane, on the line $$ y = 4x$$ The solution should be: $$T=\pmatrix{0.06&0.235\\0.235&0.94}$$ But somehow i ...
0
votes
0answers
12 views

Performing a shift on a piecewise function

Following is a convolution of $y(t) = h(t)*x(t)$ $$y(t) = \cases{\frac{1}{2}e^{2} &\text{ if } t\ge 1 \cr\cr \frac{1}{2}e^{2t} &\text{ if }0 \lt t\lt 1 \cr\cr 0 &\text{ if } t\le 0}$$ ...
1
vote
1answer
42 views

How to find an unitary transformation of $A$ that minimize $(A'_{i,i}-1)^2$?

Is there a way to find an unitary transformation $$ A'=U^+AU $$ that minimize: $$(A'_{i,i}-1)^2$$ In other words, the diagonal elements must be similar to one: $A'_{i,i} \approx 1$ Any hint? ...
0
votes
1answer
33 views

Linear transformation

Let there be a linear transformation $T:R^3\rightarrow R^2$ Is there a linear transformation so that: $Ker(T)=Span((1,2,1),(0,3,-1))$ and $Im(T)=Span((5,-7))$ Answer: $Dim(V)=Rank(T)+Null(T)=2+1=3$ ...
0
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0answers
12 views

Does logged data have to be transformed back to its original form before testing the accuracy?

I have run my data through a model in r, i ran ARIMA to forecast. The model forces a log transformation to be applied to the data. To test the accuracy of the fitted model formed by ARIMA would i need ...
0
votes
1answer
32 views

Why does the discrete cosine transform compact the information at the “low frequencies”?

I've been investigating about the discrete cosine transform. I think I understand the practical applications it has and how it is used in image/audio compression. I also know it is related with the ...
0
votes
0answers
10 views

Partial derivative of exponential of cosine transform of a vector

I am looking at the following partial derivative result that involves a cosine transform. Please refer to equation (6) and (9) in ("A unified framework of HMM adaptation with joint compensation of ...
1
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1answer
45 views

How do you prove a hilbert transform?

I am stuck with this question below, I need help;
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2answers
32 views

Writing a composite transformation as a matrix multiplication

I am confused about a question on matrix multiplication of a transformation. I have two matrices, P and Q as follows: $$P = \begin{pmatrix}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} ...
0
votes
0answers
36 views

Reflection about y=-5

What would be the reflection matrix when reflected about line y=-5. E.g. line segement with endpoints (-1,3) & (6,-2) will become a line segment with endpoints (-1,-13) & (6,-8). I have tried ...
0
votes
0answers
27 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
1
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2answers
58 views

How to find the equation of the graph reflected about a line?

Consider the graph of $y = e^x$ (a) Find the equation of the graph that results from reflecting about the line $y = 4$. (b) Find the equation of the graph that results from reflecting ...
1
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0answers
31 views

Show that a transform is involutive

Let $\mathcal C$ be the class of continuous, nonnegative, not identically equal to zero, concave, positive homogeneous of first order functions from $\mathbb R^n_+ = \{ x \in \mathbb R^n \colon x \geq ...
0
votes
1answer
24 views

Transformation of a surface normal

I'm taking a university level course in discrete geometrics and graphical programming, and I'm having trouble understanding this exercise. Let p be a point in R^3, n a surface normal, and M a ...