1
vote
0answers
19 views

Geometric meaning ov VO with O unitary matrix [closed]

What is the geometric meaning of VO where V is a n nxn Matrix and O is a unitary matrix?
0
votes
0answers
38 views

Why does distance lose meaning in high-dimensional space?

I'm working on an algorithm that clusters points in extremely high-dimensional space (thousands, if not more). However, I came across this wikipedia page: ...
0
votes
0answers
11 views

Find the reference point required to transform scale two elements uniformly

This is actually a programming issue I am having but the answer is rooted in matrix mathematics so this seems like the best place to ask it. I am no mathematician so I apologise if some of my concepts ...
0
votes
2answers
14 views

Projection matrix to project a point in a plane

How to determinate the 4x4 S matrix so that the P gets projected into Q, on the XZ (Y=0) plane? Q = S P
0
votes
4answers
194 views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
1
vote
1answer
27 views

Calculating the adjustment translation to be applied after rotating and scaling so that operations pivot about a given point.

I have a matrix for transforming an image into a target frame. The matrix is a function of a scale, $s$ rotation angle, $\theta$, and a translation that is applied after rotating, $tx, ty$. The ...
1
vote
0answers
45 views

Perspective projection alternate matrix (SOLVED)

A lot of perspective projection matrices I've seen look something like this: $$\left [\matrix {1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&z&0}\right]$$ where $z$ is ...
4
votes
4answers
130 views

rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
0
votes
1answer
50 views

Inner product space, orthonormal bases and change of basis.

I define unitary as $B*B=I$ I know that part (i) requires me to show the matrix coefficients are that of the inner product for bases A and B, however I am unsure how to get to this. Any help would ...
0
votes
1answer
10 views

Symmetric Parallelograms Under Linear Transfer Marticies

I am trying to show that a parallelogram which is symmetric about the origin stays symmetric about the origin under the action of a linear transfer matrix. It is a fairly trivial case to draw a ...
2
votes
1answer
47 views

Classification of parabolic elements of a subgroup of $PSL_2(\mathbb R)$

Let $G\subset PSL_2(\mathbb R)$ be the group generated by the matrices $$a_n=\begin{pmatrix} 1 & 2\cot\frac{\pi}{n}\\0 & 1\end{pmatrix},\; c_n = \begin{pmatrix} ...
1
vote
1answer
27 views

From world space to object's space. Scaling.

I am developing a ray tracer and I need to compute intersections between many surfaces and rays. A classical method to make the computation time lower and the code simpler is to define some constants ...
1
vote
0answers
35 views

What does it mean to compute a normal to a triangle in a “clockwise direction”

I am trying to understand how this works. I am given 3 points, each representing a vertex of a triangle. I must then "organise" the points and calculate the normal of the resulting triangle in a ...
1
vote
0answers
26 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
1
vote
2answers
102 views

Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...
0
votes
1answer
28 views

Matrix Transformation

Consider a triangle that has vertices at A(1,2), B(2,3) and C(4,2). Reflect this triangle in the line through the origin which is inclined at 30 degree to the positive x-axis. Find the vertices of the ...
1
vote
1answer
51 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
2
votes
1answer
78 views

Geometric interpretation of matrices

I'm interested in knowing some geometric interpretation of matrices. Can you suggest any lecture note or textbook or anything else about it? I've just finished an undergraduate course in linear ...
4
votes
0answers
46 views

What does the metric matrix G tell us here

Let $\phi:U \rightarrow S \subseteq \mathbb{R}^3$ be a chart from $U \subseteq \mathbb{R}^2$ to a surface $S$. $G = g_{ij}$ be the metric matrix such that $ g_{ij} = \frac{\partial \phi}{\partial ...
0
votes
2answers
66 views

Find all $2\times 2$ orthogonal matrices $A$ such that $2A^3 = B$

I need to find all possible $2 \times 2$ orthogonal matrices $A$ such that $$ 2A^3 = \begin{pmatrix} 1 & -\sqrt{3}\\ \sqrt{3} & 1 \end{pmatrix} $$ I have thought of the given matrix, divided ...
16
votes
0answers
413 views
+50

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
0
votes
0answers
25 views

Find angle-preserving transformation matrix given 2 points

I asked a similar question yesterday about finding an affine transform matrix given the same 2 points in both coordinate systems. I was told that there was only a unique solution, if the scaling was ...
0
votes
1answer
107 views

Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
1
vote
1answer
74 views

Why do lattice cubes in odd dimensions have integer edge lengths?

This is a spinoff from Characterization of Volumes of Lattice Cubes. That question claims a number of facts as being proven, but doesn't include the full proofs. That's fine for the question as it ...
11
votes
1answer
203 views

Characterization of Volumes of Lattice Cubes

Here is a problem that came up in a conversation with a professor. I do not know if he knew the answer (and told me none of it) and has since passed so I can no longer ask him about it. Let $C$ be a ...
7
votes
4answers
178 views

Geometry of the Cayley Transform

I'm trying to understand the geometry of the Cayley transform. Suppose I have a $3 \times 3$ rotation matrix $R$ (i.e an orthogonal matrix with determinant equal to $1$). Let's ignore the corner case ...
0
votes
1answer
45 views

Describing Cartesian transformations in Cylindrical Polar Coordinates

I have a question about converting functions defined in Cartesian coordinates to a cylindrical polar system. The particular coordinate transformation that I'm reading about is: \begin{equation} ...
0
votes
1answer
53 views

Projection Matrix between two Vectors

Given a two normal vectors v1 = [a1;b1;c1] and v2 = [a2;b2;c2] as given in Fig1. How I can derive the projection matrix that ...
2
votes
1answer
241 views

Eigenvalues of 3x3 Covariance Matrix, Geometric Interpretation

Problem Definition I would like to code an algorithm for decomposing a covariance matrix into its eigensolution (set of eigenvalues and corresponding eigenvectors. In my specific case I want to deal ...
1
vote
1answer
97 views

Rotate vector using transformation matrix, and read some angle

I need to rotate a vector using transformation matrix. For example: I heave vector Z (0, 0, 1). I'm rotating it by 100 deg around Z-Axis. Result will be the same as input. How to compute the angle ...
0
votes
2answers
107 views

How to find plane of reflection from transformation matrix

If you have an orthogonal matrix with a determinant of -1, how do you determine the plane of reflection? Thanks
-1
votes
2answers
73 views

$\alpha^{-1}(\ker(\beta))$, how to find? [closed]

I can't understand how to find $$ \alpha^{-1} (\ker(\beta)) $$ where: $$ \alpha = \pmatrix{1 & 2 & 1\\0 & 1 & 0}\\ \beta = \pmatrix{0 & 1\\ 0 & 1 } $$
0
votes
1answer
65 views

Elliptical polarisation

In physic context one find the curve with parametrisation in t, $x=x_0\cos(t)$ and $y=y_0\cos(t+\varphi)$ with is an ellipse with equation ...
1
vote
1answer
219 views

Calculating angle of rotation of orthogonal 3x3 matrix

Regarding the matrix in Q3b here: http://www.maths.ox.ac.uk/system/files/coursematerial/2013/2637/5/13sh2.pdf I've worked out the axis of rotation by finding out the line of invariant points, but I'm ...
2
votes
2answers
66 views

Equation of plane without cross product

We know that vectors $(3,3,4)$ and $(-1,-1,5)$ span a plane in $\mathbb{R}^3$. Can we somehow readily infer that the plane's equation is $x_1 - x_2 = 0$? Cross-products have not yet been introduced ...
1
vote
1answer
82 views

Constructing a rotation matrix from complex eigenvalues

I am trying to construct a rotation matrix $\mathbf{R}\in\mathbb{R}^{3\times3}$ rotating around an axis $\hat{n}$ in a basis $\{\hat{n},\hat{u}_{1},\hat{u}_{2}\}$. Formally: Given a basis ...
0
votes
3answers
48 views

Getting $x,y$ position on an image based on given value

This should be simple but my math skills are really bad ... I have an image of 36 images (6 by 6 matrix). These small images are 36 instances of a direction arrow (like from Google maps GPS), each ...
18
votes
3answers
707 views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
2
votes
0answers
68 views

What would this set look like

Let $S\subseteq\mathbb{R}^{3}$ be the set of $\left(x,y,z\right)$, $x\ge y\ge z$ , which are the three eigenvalues of $diag\left(1,2,3\right)+Udiag\left(-1,-2,-4\right)U^{T}$, where $U$ is an ...
1
vote
1answer
125 views

Finding the “differentness” of two point clouds

I would like to reduce the "differentness" of two point clouds $X$ and $Y$ to a single comparable value $\lambda$, which would ideally be $0$ when $X$ and $Y$ are identical upto isometry (rotation, ...
2
votes
2answers
64 views

Give a geometric interpretation of | I | = 1 for I the identity matrix.

Can anyone help me in giving a geometric interpretation of | I | = 1 for I the identity matrix.?
2
votes
2answers
243 views

Are there non-affine matrices?

Matrices are useful for proving statements like The ratio between the areas of a parallelogram and the quadrilateral formed by joining their midpoints is $2$. The ratio between the volumes ...
3
votes
2answers
598 views

Find the eigenvalues and eigenvectors of A geometrically

I am really confused with this question: Find the eigenvalues and eigenvectors of A geometrically: $$ A = \begin {pmatrix} 0 & 1 \\ 1 & 0 \end {pmatrix} $$^ reflection in the line $y=x$. ...
0
votes
1answer
151 views

Direction Cosines and Rotation Angles

I'm rotating an object in 3D space with respect to a relative base, or reference frame. I'm using a normal vector to represent the rotation angles. Suppose you have an object parallel to the ...
1
vote
0answers
241 views

What is the modern use of $\bigodot$ sign?

I've seen $\bigodot$ used in various contexts. It's used for a special set theory operation by some authors (say, Saks) and as sign for Hadamard product by a couple other authors (say, Wiener) in the ...
0
votes
1answer
60 views

Checking understanding of concept

I want to check if I have understood a concept correctly. Problem: Describe geometrically the action of an orthogonal $3$ x $3$ matrix with determinant -1. My solution: The orthogonal $3$ x $3$ ...
1
vote
1answer
140 views

Perpendicular unit vectors

I have a known unit vector $p (a,b,c)$. First I want a unit vector $q$ which is perpendicular to $p$ and passing through a known point $V(X_0,Y_0,Z_0)$. Then a another unit vector $r$ which ...
0
votes
0answers
21 views

How to set dihedral values to null?

I have a protein with many residues, but I would like to set the phi and psi angles of residue 15 to value of null. I have a file containing all residues and Cartesian coordinates, and I have another ...
1
vote
0answers
2k views

How to compute homography matrix H from corresponding points (2d-2d planar Homography)

I went through this thread Mapping Irregular Quadrilateral to a Rectangle If i know the 4 corresponding points in image say p1->p1' p2->p2' p3->p3' p4->p4' then how to compute pi(x,y) from pi'(x,y) ...
0
votes
0answers
40 views

Can you compute the location that a camera is point at?

I have a GoPro camera mounted under a RC plane. The plane is at some latitude, longitude and altitude as well as having (roll, pitch and yaw). Also the camera is on a platform that has it's own ...