0
votes
1answer
21 views

Linear equation so that the solutions has to be integers.

I have this equation $$-X_5-32X_4-32^2X_3-32^3X_2-32^4X_1=[PARAMETER]$$ I want to get all the solutions to this equation with a given parameter(integer) and all of the solutions has to be integers ...
0
votes
1answer
38 views

Is there any algorithm (if possible, I need the codes) for Jordan normal form decomposition for large matrices in practice?

Although it is an ill-posed problem as B Kågström said in "An algorithm for numerical computation of the Jordan normal form of a complex matrix", I wonder what people do when they need to do Jordan ...
3
votes
0answers
76 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
0
votes
1answer
68 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
1
vote
1answer
38 views

Why we take transpose of Vector (Displacement Vector)?

I'm trying to understand some equations that involves transpose of vectors (displacement vectors to be precise) Two set of vectors F and G (with i,j) that corresponds to X,Y value in plane and ...
0
votes
0answers
40 views

It would be possible to use the covariance matrix $C'=XX^T$ instead of the standard $C=X^TX$ to get the same result on PCA?

It would be possible to use the covariance matrix $C'=XX^T$ instead of the standard $C=X^TX$ to get the same result on PCA? If so, what are the next steps to retrieve the data with reduced ...
1
vote
1answer
68 views

texture mapping from a camera image to a 3D surface acquired by a kinect

I have the following problem: A kinect camera capture a 3D surface and save it as a .obj files containing all the positions of the vertices (in the kinect coordinate system). If I take a picture ...
3
votes
1answer
113 views

How to solve this system of 3 equations with 3 variables?

I stumbled upon this system with constants $a_{i,j}>0$ that I want to solve for $x,y,z \in\mathbb{R}$: \begin{align} a_{2,1}y+a_{3,1}z=& x(y+z) \\ a_{1,2}x+a_{3,2}z=& y(x+z) \\ ...
1
vote
0answers
49 views

Computationnal geometry: vector, basis, point and coordinate system?

I am trying to build a small geometrical library in C++, that is mathematically consistent (not so false). The goal here is to construct two concepts: vectors and points. I am not sure that the ...
2
votes
2answers
96 views

Avoid dividing by zero with just variables and basic operators

I am working on stats for a sports team, and one of the stats I have the ratio of Shots and Shots on Target (Which I call ...
0
votes
0answers
60 views

Inward-pointing normal and co-ordinate systems

I'm doing a course in computer graphics, and as such, we're being taught measures on how to deal with the Hidden Surface Removal problem. One of the topics covered was "back-face detection", that is, ...
8
votes
0answers
133 views

Algorithm for obtaining the surface of a mirror

My colleague and I have been trying to implement an algorithm described in the paper "Recovering local shape of a mirror surface from reflection of a regular grid", primary author of which being ...
0
votes
2answers
54 views

Representing everywhere a camera can see as a matrix

I'm learning about Computer Graphics and there is one point really puzzling me. I understand that vertices (vectors) represent points in space and that transformation matrices represent changes that ...
3
votes
2answers
578 views

Learning Proofs (for Computer Science)

Harvard's math curriculum, for freshmen, is divided into 4 classes beyond the BC Calculus level, Math 21, 23, 25 and 55. Math 21 is your classic plug-and-chug multivariable calculus and linear algebra ...
0
votes
1answer
67 views

Can a directed hamiltonian path be found in polynomial time?

I was discussing a programming competition problem with one of my math professors in Linear Algebra that reads as follows: A matrix is an $r\times c$ array of numbers, where $r$ is the number of ...
1
vote
2answers
41 views

Solving system of linear eqaution in special cases

I have to solve for $Ax=B$. Here the diagonal elements of $A$ are $-1$ and all other elements are $1$. $A$ is $n \times n$ matrix . In this special case can we solve for $x$ quickly? EDIT: quick is ...
0
votes
1answer
60 views

Algorithm to check if a matrix is elementary

I'm currently writing a homework problem for a linear algebra course and I'm trying to come up with an algorithm to check if a matrix is elementary. That is, to check if it is one of the three forms ...
0
votes
0answers
44 views

Deriving the fundamental equation relationship

I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic ...
5
votes
1answer
120 views

Maximal subset with rank $k$

I'm trying to solve the following problem for an algorithm I'm trying to develop and I couldn't find anything helpful in scholar google. Here is the question: Suppose I have a set of $N$ vectors ...
0
votes
1answer
43 views

Difficulty understanding some algebra done in a problem

The main problem is about computer science, trying to show that $f(x)=e^{x^Tx'}$ is of the form $\exp{\Big( \frac{||x - x'||^2}{2\sigma^2} \Big) }$, so it could be a kernel function. (see here for ...
1
vote
4answers
83 views

Procedures to find solution to $a_1x_1+\cdots+a_nx_n = 0$

Suppose that $x_1, \dots,x_n$ are given as an input. Then we want to find $a_1,\ldots,a_n$ that satisfy $a_1x_1 + a_2x_2+a_3x_3 + a_4x_4+\cdots +a_nx_n =0$. (including the case where such $a$ set does ...
3
votes
3answers
219 views

0-1 knapsack like - the set of all non-contained affordable binary selections

This is my first question here, so please go easy on me :) The following problem is – I think - similar to the 0-1 knapsack problem. It's simplified somehow in that each item has only a cost ...
1
vote
0answers
142 views

Question about the elementary divisors of a special matrix

I have the following question: Is there a closed formula for the elementary divisors of the Matrix $M={(m_{ij})}_{i=1,...,n,\ j=1,...,k}$, where ${m}_{ij}$ is the greates common divisor of $i$ and ...
0
votes
1answer
150 views

Operations it takes to invert a matrix

I had this question on a homework and I was wondering if my thinking was correct. if you have a nxn matrix, how many total arithmetic operations do you perform? ( +,-, *, /)...I went about this ...
4
votes
1answer
272 views

Matrix Chain Multiplication?

The following are questions about using dynamic programming for matrix chain multiplication. Pseudocode can be found in the Wikipedia article on matrix chain multiplication. 1) Why is the time ...
5
votes
2answers
379 views

Distance between a point and a m-dimensional space in n-dimensional space ($m<n$)

I am trying to find a method with a low computational cost to compute the distance of a point $P$ and a space $S$ that is defined by the origin $O$ and $m$ vectors $v_1, v_2, ..., v_m$ in an ...
2
votes
3answers
5k views

Inverse of transformation matrix

I am preparing for a computer 3D graphics test and have a sample question which I am unable to solve. The question is as follows: For the following 3D transfromation matrix M, find its inverse. Note ...
1
vote
2answers
258 views

Understanding Matrix Formula with Scant Knowledge of Linear Algebra

$n$ is a power of $2$. $M =\pmatrix{ 1& x_0 & x_0^2 & \dots &x_0^{n-1}\\\ 1& x_1 & x_1^2 & \dots &x_1^{n-1}\\&& \vdots\\1& x_{n-1} & x_{n-1}^{2} ...
2
votes
0answers
85 views

Need little hint to prove a theorem .

I have an iterative method \begin{eqnarray} X_{k+1}=(1+\beta)X_k-\beta X_k A X_k~~~~~~~~~~~~~~~~~ k = 0,1,\ldots \end{eqnarray} with initial approximation $X_0 = \beta A^*$ ($\beta$ is scalar ...
2
votes
1answer
122 views

Security analysis of a matrix multiplication protocol

Suppose Alice would like to obtain the product of two $m\times m$ matrices i.e. $A$ and $B.$ Alice has $A,$ whereas Bob has $B.$ Since Alice does not want to reveal $A$ to Bob, she chooses a ...
0
votes
0answers
97 views

Convert triangular real matrix to hermitian

We are developing some computer program which at some point uses a library (for which we do not have access to its source code) to solve the general eigenvalue problem; given two input real symmetric ...
4
votes
2answers
3k views

Practical applications of eigenvalues/eigenvectors in computer science

What are the most important/popular applications of eigenvalues and eigenvectors in practical terms, in fields such as computer science and computer graphics? Wikipedia does mention some but doesn't ...
5
votes
2answers
2k views

Computational complexity of computing the determinant

The formula for the determinant of an $n$ by $ n$ matrix given by expansion of minors involves $n!$ terms. As such, computing the determinant of a given matrix of with integer entries via expansion by ...
5
votes
1answer
348 views

Rank of an interesting matrix

Lets define: $U=\left \{ u_j\right \} , 1 \leq j\leq N= 2^{L},$ the set of all different binary sequences of length $L$. $V=\left \{ v_i\right \} , 1 \leq i\leq M=\binom{L}{k}2^{k},$ the set of ...
8
votes
3answers
3k views

fast algorithm for solving system of linear equations

I have a system of linear equations, $Ax=b$, with $N$ equations and $N$ unknowns ($N$ large) If I am interested in the solution for only one of the unknowns, what are the best approaches? for ...
0
votes
1answer
212 views

Deriving an expression for an epipolar line

I would like to derive an expression for an Epipolar Line that appears in a right camera (there are two cameras - a left a right one). These cameras are placed in a $(x,y,z)$ vector space. This is a ...
8
votes
1answer
500 views

Incremental calculation of inverse of a matrix

Does there exist a fast way to calculate the inverse of an $N \times N$ matrix, if we know the inverse of the $(N-1) \times (N-1)$ sub-matrix? For example, if $A$ is a $1000 \times 1000$ invertible ...
1
vote
1answer
203 views

Rank of a graph matrix

$G$ is a bipartite graph with $2m$ nodes on the left $(u_0..u_{2m-1})$, and $2^{m}$ nodes on the right $(v_0..v_{2^{m}-1})$. There is an edge (connection) between $u_i$ and $v_j$ iff $(i+1)$'th ...
3
votes
1answer
93 views

An efficient way to check whether a polynomial (under certain condition) is absolutely equal to zero or not

We have a function $f$ of $N$ variables which is the product of $M$ polynomials: $$f(x_1,x_2,\ldots, x_N) = P_1 \cdot P_2 \cdots P_M.$$ Each $P_i$ is a polynomial of at most three variables ...
5
votes
3answers
622 views

knapsack algorithm that looks too good to be true

I have an idea for solving the knapsack problem, but it looks too good to be true. I would like someone to explain potential problems with this approach. I'll give an example: I want to find a subset ...
5
votes
2answers
518 views

What is the best way to factor arbitrary polynomials

I am currently working on a Computer Algebra System and was wondering for suggestions on methods of finding roots/factors of polynomials. I am currently using the Numerical Durand-Kerner method but ...
6
votes
3answers
1k views

What is linear programming?

I asked this question on Stack Overflow but it was closed as "not programming related". So I think this is probably the best place for it... I read over the wikipedia article, but it seems to be ...