Articles of transformation

Algorithm to determine matrix equivalence

I’m a physicist who’s not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I’m just unaware of. I have two real rectangular matrices $A_{mxn} B_{mxn},$ where $m>n$ whose entries are $\pm1,0$. As a concrete example, take $$ A = \begin{bmatrix} 1 & 0 & 0 \\ […]

Consider the trace map $M_n (\mathbb{R}) \to \mathbb{R}$. What is its kernel?

The map is the trace map. I.e, it takes any $n$ by $n$ matrix and associates to that matrix, a number of the form $\mathrm{Tr}(A) = \sum_{i=1}^n a_{ii}$, where $A \in M_n (\mathbb{R})$. I need to find the kernel of this map, give a basis and its dimension (which is easy once I have the […]

Householder reflections

Let $x=\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero but I’m doing something wrong… $U=I-\frac{uu^T}{\beta}$ $\beta=\frac{\left \| u \right \|_2^2}{2}$ $Ux=x-u$ $\beta=\frac{16}{2}=8$ $u=\begin{bmatrix} 0\\ 2\\ 3 \end{bmatrix}$ $U=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & […]

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three different matrices in $\Bbb{R}^2$ and sometimes in $\Bbb{C}^2$ which represent a part of the control system and each matrix has just two […]

Find rotation that maps a point to its target

I have a 3D point that is rotated about the $x$-axis and after that about the $y$-axis. I know the result of this transformation. Is there an analytical way to compute the rotation angles? $$ v’=R_y(\beta)*R_x(\alpha)*v $$ Here, $v$ and $v’$ are known and I want to compute $\alpha$ and $\beta$. $R_x$ and $R_y$ are […]

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. Therefore the matrix must be $ \begin{pmatrix} 1&1\\1&-1 \end{pmatrix} $. But in maple it is just drawn as a straight line. However, it should be a rotated hyperbola. Is […]

I'm looking for the name of a transform that does the following (example images included)

I’m in the usual situation that if I would know what the name of the thing was, then I could find the answer. Since I dont know the name, here is what I’m looking for: Suppose I have the following “snake” of 10 quadrilaterals: I now want to apply a transformation to each of these […]

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I’m looking for a “best fit” transformation consisting of a rotation and a translation. I.e. $$ \min_{A,b} \sum (A p_i + b – q_i)^2 , […]

Does the Fourier Transform exist for f(t) = 1/t?

My professor says that the following function has a Fourier Transform: $$f(t) = \frac{1}{\pi t}$$ He said that all I have to do is apply some of the Fourier Transform properties and not the direct integral definition of the Fourier Transform to find it: However, My book for the class claims that no Fourier Transform […]

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 – 36 x y^2 + 38 y^3 – 342 x y z + 96 y^2 z – 144 x z^2 + 456 y z^2 + 1316 z^3 = […]