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

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matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
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2answers
22 views

Finding only first row in a matrix inverse

Let's say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what's the best algorithm to use to calculate just this row? ...
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12 views

Solve set of poorly conditioned linear equations in block matrix form

I would like to solve the following set of linear equations where A, B, C and D are each 4x4 matrices. K is then an 8x8 matrix The values in A and D have magnitudes of $\approx 10^{17}$, B has ...
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20 views

Write down a linear programming problem

I want to replicate a linear programming problem.I have the following information, for the background." A fuzzy regression analysis with only one independent variable X results in the following ...
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0answers
26 views

Lagrange multiplier for more than one constraints.

How to minimize $x^TAx$ over the set $D=(x\geq 0, x^TBx=1$ and $(I-A^\dagger A)x=0$), where $A$ is copositive matrix of order $n-1$ and $B$ is strictly copositive matrix of order $n$. If I drop the ...
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1answer
9 views

R3 to Planar Subspace Tranform

I'll ask my question three ways to try to maximize my chances of successful communication. I have: a point 'P' in R3 with coordinates $(P_X,P_Y,P_Z)$ A plane Defined by: A point 'O' given by ...
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0answers
15 views

Properties of Lagrange Interpolation [on hold]

Using Lagrange to show that (a) $\sum x^k l_j(x)=x^k$ and (b) $\sum (x_j-x^k)\cdot l_k(x)=0$
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26 views

Solve linear system with matlab

In my problem, $A$ is a $m \times n$ matrix with $m \geq n$ and $\mathrm{rank}(A)= n$. Let $\Gamma$ be the $(m+n) \times n$ matrix defined by : $$ \Gamma = \begin{bmatrix} A \\ \mathrm{I_{n}} ...
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1answer
39 views

Generalized inverse/Pseudo Inverse

Let $A_{m. n}$ be a matrix with rank $p$ where $p\leq m$ and $p\leq n$. First Question: We need to show that $A$ can be decomposed as a product of two matrices $A=BC$ where $B$ is an $m$ by $p$ and ...
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0answers
16 views

computing leftmost eigenpair of positive-definite matrix

Let $A$ be an $n\times n$ real symmetric positive-definite matrix. Assume that $n$ is large and that $A$ is dense (i.e. it is not sparse). Question: What is the state-of-the-art algorithmically for ...
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0answers
36 views

Test for powers method

I have been told that for a normal matrix $A$, the powers method (i.e. computing the succession of Rayleigh quotients for a succession of vectors $z_k=A\cdot z_{k-1}$) can use the following stop ...
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1answer
39 views

Inverse of a diagonal matrix plus a constant

I am looking for an efficient solution for inverting a matrix in the following form: $D+aP$ where D is a (full-rank) diagonal matrix, a is a constant, and P is a one matrix. This question Inverse of ...
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2answers
118 views

How to efficiently solve a series of similar matrix equations using the LU decomposition

This is the problem I'm dealing with: Let $\sigma_1,\dots,\sigma_n \in \mathbb{R}$ and $b_1,\dots,b_n$ be column vectors of length $n$. Consider the system $$ (A - \sigma_jI)x_j = b_j, \quad ...
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1answer
23 views

Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...
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24 views

Are there any sure-fire methods for correctly arranging matricies for Gaussian Elimination?

I am attempting to make a Gaussian Elimination solver for systems of linear equations that contain less than 100 equations. I have roughed out a method for creating and filling in the diagonal of a ...
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4answers
83 views

Shortest Distance between a Point and a Numerical 2D Curve

I have a 2D Curve. I have all the numerical values for the line within a certain range. I do not have an equation for this line. At several points in this 2D space I want to calculate the shortest ...
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1answer
27 views

Equation that maps two values to three? [closed]

I'm trying to get an equation for a line in 2-dimensions that maps into an equation for a line in 3-dimensions where $$ f(x,y) => x_1,y_1,z_1 $$ and preferably have a resource that teaches me the ...
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15 views

Stuck on condition number derivation of the perturbed equation $(A + \Delta)\tilde{x} = b + \delta_b$

I've almost got what I want. We start with $Ax = b $ and $(A + \Delta)\tilde{x} = b + \delta_b$. What I have then is \begin{align*} \tilde{x} - x &= -A^{-1}\Delta\tilde{x} + A^{-1}\delta_b \\ ...
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0answers
39 views

compute the bisecting normal hyperplane between two $n$-dimensional points.

I have two points $\mathbf{x_1}$ and $\mathbf{x_2}$, where $\mathbf{x_i}=\{x^i_1, x^i_2, \ldots, x^i_n\}$. I need to find a normal hyperplane $P$ that goes through the midpoint of $\mathbf{x_1}$ and ...
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1answer
32 views

Jacobi vs. Gauss-Seidel: convergence

I know that for tridiagonal matrices the two iterative methods for linear system solving, the Gauss-Seidel method and the Jacobi one, either both converge or neither converges, and the Gauss-Seidel ...
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1answer
26 views

Tridiagonalize matrices with Householder transformation

I know that it is possible to tridiagonalize symmetric matrices by using a Householder trafo. I also found that we can get any matrix to Hessenberg form by using Householder trafos, but I still don't ...
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0answers
33 views

Integration of ODE equation in Matlab / Octave

I have a system of 8 ODE's where the initial conditions are in matrix form. $\frac{dT}{dS} = H T$ where T at the initial state is the identity matrix. $T(a) = I$ H is a constant 8x8 matrix T is ...
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0answers
14 views

Is the Hessenberg form of a matrix unique?

I have to calculate the Hessenberg form of an matrix using householder reflectors. For real Matrices I get the same result as the 'hess()' function in Matlab, but for Complex Matrices I keep getting ...
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52 views

Alternative to the Gram-Schmidt Procedure for Orthogonalization

I was wondering if there is an alternative to the Gram-Schmidt procedure, which instead of being a successive orthogonalization scheme, would be non-successive (simultaneous)? In other words, is there ...
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1answer
17 views

Matrices admit a QR decomposition

I just wanted to ask which matrices admit a QR decomposition. I think that all matrices $A \in \mathbb{R}^{m \times n}$ with $m \ge n$ admit a QR decomp. Are these the only ones that have a QR decomp, ...
4
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1answer
187 views

Generate arbitrary numerically invertable matrix

I'm designing a unit-test for a matrix inversion function. Currently I make a random matrix as a test case by generating its elements with random numbers uniformly distributed in $[0,1)$. If I ...
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0answers
30 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: ...
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0answers
52 views

Proof of theorem about iterative methods

How do I prove that if $A$ is a tridiagonal (or block tridiagonal) matrix then the corresponding $P_J$ and $P_G$ iteration matrices for the Jacobi and Gauss-Seidel methods satisfy that if $\lambda$ is ...
3
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1answer
68 views

3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
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1answer
15 views

Condition number of a matrix bounded from below and above?

Is condition number of an invertible matrix bounded from below? And is condition number always bounded from above for an invertible matrix?
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1answer
31 views

Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
0
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1answer
30 views

Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge b$$ Where Q is square ($n$x$n$), positive semi ...
0
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2answers
98 views

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
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12 views

How is this a substitution? Linear algebra transformation matrix misunderstanding

I found the following matrix equation in '3D Surveillance System Using Multiple Cameras', (authors: Ajay Kumar Mishra, Bingbing Ni, Stefan Winkler, Ashraf Kassima) (link here): I don't follow the ...
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0answers
10 views

find the mean of a collection of the solutions of given linear systems

Suppose one has a group of $k\times k$ square matrices $A_i, i = 1,2..n$, another $k$-vector $b$ is also given. I want to exam the mean of the group of vectors $x_i$ over $i = 1,2...n$ where $A_i * ...
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1answer
67 views

Finding the smallest max eigenvalues for related matrices?

While messing around with a spectral approach to a graph coloring question, I happened upon a type of problem I hadn't seen before. Suppose you have two symmetric $n$ x $n$ matrices in the form ...
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1answer
46 views

Unique least square solutions

There is a theorem in my book that states: If $A$ is $m\times n$, then the equation $Ax = b$ has a unique least square solution for each $b$ in $\mathbb{R}^m$. But can we find a counter-example to ...
4
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1answer
29 views

Implementing trig functions for dual numbers

I'm curious, how do common trig functions get implemented for dual numbers? One way would be to use the power series definition, but that seems inefficient
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0answers
21 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
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0answers
12 views

Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute ...
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0answers
17 views

Nearest points / residuals on a total least squares parabola

Consider fitting a parabola $y = a + bx + cx^2$ to 2d data $X_i, Y_i$ with noise in both X and Y, using the the singular value decomposition as in Total_least_squares (TLS): $\qquad X = [ 1\ \ Xdata\ ...
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0answers
13 views

Transformation between Ideal and Warped Surface

I work on manufacturing metal panels with holes drilled in them. Suppose I have an ideal 3D surface from CAD. I want to compare it to the actual part using reference points to compare between the two. ...
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4 views

How can I apply a median filter directly to a time-varying rotation matrix?

I need MatLab script which would take a series of rotation matrices (referring to an actual physical object's orientation) and apply median filter to it to eliminate speckle noise from it. The way ...
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0answers
64 views

Numerically approximate the maximum of an element of a vector after a series of matrix multiplications.

Where S is a sigmoidal function, A_i is a matrix, and x is an input vector, and ...
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0answers
40 views

Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...
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0answers
39 views

Complexity of the power method

I'd like to find out what the complexity of the power method is depending on the size of the matrix $A \in \mathbb{R}^{n\times n}$ given that the algorithm runs until a certain stop criterion. I.e. ...
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27 views

resources about sparse global constrainted optimization

Please recommend a good resources (books/articles/software) about sparse global constrained optimization?
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1answer
53 views

Solving 2x2 diagonally dominant matrix systems (non-symmetric)

I have a linear system of the form $Ax=b$ where $A\in \mathbb{R}^{2\times2}, b\in \mathbb{R}^{2\times1}$. A is diagonally dominant and non-symmetric. This is a "kernel" that I am using to solve a ...
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30 views

Numerically stable way to compute $\text{Trace}[\mathbf A\mathbf A_1^{-1}\mathbf B\mathbf B_1^{-1}]$

I have two dense (column-major) PSD matrices $\mathbf A$ and $\mathbf B$, $\mathbf A,\mathbf B\in\mathbb R^{n \times n}$ ($n$ is usually $\sim 1000$) and also $\mathbf A_1=\mathbf A+\eta\mathbf I_n$ ...
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1answer
47 views

Application of Conjugate Gradient Method to non-symmetric matrices

I am currently working on a problem in which I am using the Conjugate Gradient method to solve for the steady state solution of a continuous time Markov chain. I am applying the algorithm found in ...