Questions on the various algorithms used in linear algebra computations (matrix computations).

learn more… | top users | synonyms

3
votes
1answer
94 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) \\ ...
3
votes
1answer
74 views

Understanding higher order SVD

Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas. On a ...
3
votes
1answer
70 views

Computational cost, power method and page rank

When solving the PageRank problem for $n$ web pages, it is necessary to find a solution of the eigenvector equation $$(fM)*p = p,$$ where $$fM = dM + (1 - d)Z$$ $$Z =\frac{1}{n}*ee^T$$ $$e =[1, 1, ...
3
votes
1answer
201 views

How to Store a Banded Matrix by Diagonal

I'm taking a graduate level independent study course this semester in Matrix Computations. I'm not getting much support from the professor, so am turning to the excellent StackExchange community for ...
2
votes
1answer
51 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
2
votes
1answer
47 views

SOR and Gauss-Seidel Method - Confusion

Can anyone explain to me the SOR Method for finding the root(s) of a function? Its supposedly very similar to the Gauss-Seidel method. The Gauss-Seidel method, from my understanding, is similar to ...
2
votes
1answer
52 views

Numerical methods for inverting non positive definite matrices

I'm working on a PDE solver and need to invert the following matrix written in block form $\left( \begin{array}{cc} kM & -S \\ -S & M \end{array}\right) $ where M and S are the usual mass and ...
2
votes
1answer
115 views

Question on “avoidance of crossing”

In review of linear algebra I come across this phenomenon, the Google Book link is this: What I do not understand is Lax tried to persuade us that "there is another way of parametrizing these ...
1
vote
1answer
27 views

Can a tridiagonal matrix be rectangular?

My program works with tridiagonal matrices (calculates its LU decomposition) so before doing anythig, it stores the matrix in 3 vectors: the three diagonals only. So far my conclusion was, a ...
1
vote
1answer
33 views

How to solve an Optimization problem with linear as well as Quadratic constraints.

I want to solve the following problem, \begin{equation} \begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & \mathbf{x^T}\mathbf{Px} \\ & \text{subject to} & & ...
1
vote
1answer
33 views

Using the Gauss-Seidel method, will the matrix A converge

Just came back from my Numerical Analysis midterm, posting up the questions and my solutions for an estimation as to how I did. If you were to perform the Gauss-Seidel method on a matrix $A$, where ...
1
vote
1answer
46 views

Partial QR factorization to solve least squares problem

I'm trying to understand how to solve a least squares problem of the form: $$\begin{bmatrix}A& B \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = [b]$$ where I only explicitly solve for $y$ and ...
1
vote
1answer
72 views

Minimizing the Determinant

I would like to minimize the determinant of the following matrix, det(A) $A = (VV^T+\lambda I)^{-1}$ and $\lambda$ is set to be very small.
1
vote
1answer
147 views

Need matlab help to construct a numerical example for solving system of linear equation for random matrices

I am reading this paper(page 183). In this paper the iterative methods for computing some solution of the general restricted linear equations \begin{eqnarray} Ax = b, ~~~~ x\in R(A^{k})~~~~ b\in ...
1
vote
1answer
150 views

1D Schrodinger/Laplace equation via finite differences: incompatible eigenvalues

I need to solve a variant of the 1D Schrodinger's equation equation using finite differences, so I decided to play a little bit with the real-space representation of some operators. Using the ...
1
vote
1answer
49 views

Prefactoring to solve many similar linear systems

I am designing an algorithm that needs to solve many (large) linear systems of the form $$\Phi^\top D_i\Phi \vec x_i=\vec r_i,$$ where $\Phi\in\mathbb{R}^{m\times n}$ with $m>n$ is fixed. We will ...
1
vote
1answer
47 views

What is the range of this function

Let $\lambda_{1}(X)$ be the larger eigenvalue of the $2$ eigenvalues of a symmetric matrix X. For fixed real numbers $a,b,c,d$, what is the range of $\lambda_{1}\left(diag\left(a,b\right)-U\cdot ...
1
vote
1answer
254 views

How to find the Householder transformation?

Assume $x=(1,0,4,6,3,4)^T$. Find a Householder transformation and a positive number $\alpha$ such that $Hx=(1,\alpha,4,6,0,0)^T$. I'm sorry that I don't know how to start with this problem. A ...
1
vote
1answer
76 views

Eigenvalues of discretized linear integral operator

Suppose I have the following kernel operator: $Af(x) = \int_{-1}^1 K(x-y)f(y)dy$ which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
1
vote
1answer
133 views

Limiting Degrees of Freedom in 3D Point Registration

I'm search for some assistance in my application of Arun's algorithm for registration (fitting) of two 3D point sets using the Singular Value Decomposition: ...
1
vote
1answer
57 views

Help regarding a weird Matrix

Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on ...
0
votes
1answer
39 views

Numerical Linear Agebra

how to Prove the backward stability of the inner product ? ...
0
votes
1answer
68 views

Reducing a matrix to upper Hessenberg form using Householder transformations in Matlab

I have the below Matlab code based on what my professor gave me in class. The last line of this code is giving me an incompatible dimensions error. When I checked the size, I got that the matrix ...
0
votes
1answer
31 views

Householder Reflection

I am working on algorithms for SVD by first performing Householder transformation. I got my algorithm to work but I'm trying to gain a better intuition of it. My understanding is that the ...
0
votes
1answer
32 views

Is it possible to solve a system of equations comprising FFTs?

Consider the following known matrices, A, B, C and these unknown matrices X,Y, all of which comprise values in the Real domain. Also consider $F(x)$ as the *Fast Fourier Transform function* (the ...
0
votes
1answer
33 views

Inverse Square root of a rectangular matrix

I am trying to compute the inverse square root ($X^{-1/2}$) of a $n \times p$ matrix with $n > p$. I was wondering if we can compute it via SVD just as we do it for square diagonalizable matrices ...
0
votes
1answer
21 views

Residual norm for iterative scheme

Consider a linear system $A\vec{x} = \vec{b}$, where $A \in \mathbb{R}^{m\times{}m}$ is non-singular and positive definite. Given the following iteration scheme $\vec{x}^{(k+1)} = \vec{x}^{(k)} + ...
0
votes
1answer
48 views

determine whether the equation $Ax = b$ is consistent for every $b$ in $\mathbb R^m$

I have two problems, the first one is the following matrix: $$\begin{bmatrix}1 & 0\\ -2 & 1\end{bmatrix}$$ where the RREF is $$\begin{bmatrix}1&0\\0&1\end{bmatrix}$$ and where the ...
0
votes
1answer
54 views

Can a 6-arm star be convex

Please help me with the following question. Suppose that the constant level contours of some function $V:\mathbb{R}^{2} \rightarrow \mathbb{R}$ have the shape of a symmetric 6-arm star. Can such a ...
0
votes
1answer
34 views

Fast Gauss-Seidel convergence on low rank matrices

I stumbled upon the following remarkable fact when experimenting with the Gauss-Seidel iterative solver: First I construct a low-rank symmetric positive semi-definite matrix $A = M^TM$ with M a ...
0
votes
1answer
38 views

Completeness of eigenvectors of Hermitian Matrix.

How do you show that eigenvectors of a Hermitian matrix form a complete set of basis?
0
votes
1answer
42 views

Rate of Convergence of Generalized Iterative Method

Consider the generalized iterative method for finding polynomial roots: $z_{k+1}=z_k +d\frac{(1/p)^{(d-1)}(z_k)}{(1/p)^{(d)}(z_k)}$ where d is a positive integer. Note that Newton's Method is a ...
0
votes
1answer
85 views

Solving Poisson Equation Finite-difference using maple

How do I solving Poisson Equation Finite-difference using maple consider Poisson equation $$\frac{\partial^2u}{\partial x^2} (x,y)+ \frac{\partial^2u}{\partial y^2} (x,y) = x*e^y$$ $0<x<2$ ...
0
votes
1answer
14 views

Are Similar Matrices and Unitary Property related?

Recall that 2 matrices $A, B\in R^{n,n}$ are similar if there exists a matrix $P$ such that $A=P^{-1}BP$. In this case is $P$ always orthogonal?
0
votes
1answer
26 views

Set up for matrix solutions

I've haven't touched linear algebra in a while so I'm sorry if this seems simple but I did a google search and I am still confused. I have to find a solution to the following set of equations: ...
0
votes
1answer
103 views

Numerical range of a matrix contains the convex hull of the eigenvalues.

I am stuck with the following question. Question: Let $A \in \mathbb{C}^{m \times m}$ be arbitrary. Let $W(A)$ be the numerical range i.e. the set of all Rayleigh quotients of $A$ corresponding to a ...
0
votes
1answer
174 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
0
votes
1answer
49 views

Need Help! Recognizing types of errors: Truncation and Roundoff

I am a little unclear on the difference between the two. What exactly are they? As simplified as possible :) How can i recognize them and identify parts of formulas or algorithms that would give ...
0
votes
1answer
84 views

matrix positive semidefinite

If $n \times n $ matrices $A, B$ are positive semi-definite, matrices $P$ and $Q$ are $n\times p$ and $n\times q$ matrices and their column vectors are orthogonal, which is to say $$P^{T}P=I_{p\times ...
0
votes
1answer
149 views

approximating diagonal of inverse sum of low rank and diagonal matrices

I was wondering if there is any theorem or algorithm to approximate the diagonal elements of the inverse of sum of low rank symmetric positive semi-definite and non-negative diagonal matrix. Let me ...
0
votes
1answer
37 views

Simple question about equivalence of two forms of PCA as trace maximization over an implicit distribution

This may be a soft question of sorts. One formulation of principal component analysis is trace maximization: $$\arg\max_U \mathbb{E}_x \ [tr(U^Txx^TU)],$$ for $U^TU\le I$ and we assume that there is ...
0
votes
1answer
130 views

Positive linear combination of vectors to produce a positive vector

Given a list of vectors, I want a linear combination with positive coefficients that produces a final vector with only positive values (EDIT: this final vector is unknown; any positive vector is ...
0
votes
1answer
25 views

Given an M x N matrix, is there a way to produce an orthogonal set of N vectors of length M, where M < N?

Gram-Schmidt orthogonalization would only use the first M vectors to generate a basis of size M x M.
0
votes
0answers
60 views

Can this be expressed as an LLSQ problem? $||Ax - b|| = c$

I'm trying to minimize the following: $||Ax - b|| - c$ where: $A$ : $K \times M$ matrix $x$ : $M \times N$ unknowns ($M$ $N$-dimensional vectors) $b$ : $K$ $N$-dimensional vectors $c$ : $K$ ...
0
votes
0answers
107 views

Efficient principal pivots

Background I'm working on a numerical linear algebra package in C#. I'm trying to implement a variety of "principal pivoting" methods to solve optimization problems (specifically linear ...
0
votes
0answers
55 views

Diagonalising a huge matrix of symbolic objects

I have to diagonalise this HUGE $9\times 9$ matrix with symbolic entries which are made up of three independent variables. Can you please give me a reference as to how to do such a thing ...
0
votes
0answers
87 views

Does a single Gauss-Seidel iteration lead to unique coordinates?

I managed to reduce certain computational problem to the Gauss-Seidel solution of the following linear system: $$Ax=Ly,$$ where $A, L\in\mathbb{R}^{n\times n}$, and $x,y\in\mathbb{R}^{n\times 2}$ are ...
0
votes
0answers
42 views

Lower Rank Matrix

Given I have a matrix A of rank 3. I want to create a matrix of Rank 2 which is closest to A in the $ {l}_{2} $ / Frobenius norm. Let's call this matrix F. Is easy to achieve by the SVD, namely, if $ ...
0
votes
0answers
128 views

Singular value decomposition, possible property

Suppose a singular value decomposition on matrix $P\in\mathbb{R}^{n\times m}$ is given, $P=U\Sigma V^T$ with $U=[u_1,\dots, u_n]\in\mathbb{R}^{n\times n}$, $u\in\mathbb{R}^{n}$, containing the ...
0
votes
0answers
149 views

Estimating a solution with the Jacobi Method for solving Ax = b

I'm trying to understand how the Jacobi method works and would appreciate a walk-through of the method with a very very simple example. In particular, I don't fully understand how one goes from the ...