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

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Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
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eigenvalues for symmetric and non-symmetric matrices

I know the Power methods and Jacobi methods are suitable to finding eigenvalues for symmetric matrices, please tell me other methods for this matrices. And what are the methods for the Non-symmetric ...
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191 views

Space spanned by matrices

I have a set of 5 by 5 matrices, M1,M2,...,M19 ,M20. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should approach the ...
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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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Is this Gram-Scmidt (or an application of) it?

I am given a $2\times 2$ matrix $$\left[ \begin{array}{ccc} a & 0 \\ 0 & b \\ \end{array} \right] $$ where $a,b \in \mathbb{R}$. I was told than an orthnormal basis for the colums of this ...
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Minimum of Maximum of Addition of two vectors/arrays

Suppose you have two arrays and you want to compute the maximum of the addition of the two arrays. Now you move the second array one field to the right. Now you can compute the maximum again of the ...
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289 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 ...
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57 views

How to understand or show this?

We have $$F=ABh^{p_1}+\theta (h^{p_2})$$ $$G=Ah^{p_1}+\theta (h^{p_2})$$ We $A$,$B$ are real numbers, $h$ positive, $|h|\leq 1$ , $p_1<p_2$ natural numbers and $\theta(h)$ means that it is of ...
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Efficiently updating a vector

What is the most efficient way to make this linear algebra computation? I am interested in computing a vector $y^{(k)}$ that updates as shown below. $$y^{(k)} = A^k B A^k x$$ where the matrices $A,B ...
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60 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 ...
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65 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 ...
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Completeness of eigenvectors of Hermitian Matrix.

How do you show that eigenvectors of a Hermitian matrix form a complete set of basis?
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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 ...
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106 views

Solving Poisson Equation Finite-difference using maple

How do I solving Poisson Equation Finite-difference using maple consider Poisson equation $$\frac{\partial^2u}{\partial x^2} (x,y)+ \frac{\partial^2u}{\partial y^2} (x,y) = x*e^y$$ $0<x<2$ ...
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16 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?
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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: ...
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138 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 ...
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250 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 ...
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58 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 ...
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86 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 ...
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44 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 ...
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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.
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Matrix existence..

How to prove that for any matrix $A\in \mathbb R^{m\times n}$ ($m\geq n$) such that $rank(A)=r$ there exists a nonsingular matrix $P$ and an orthogonal matrix $U$ such that, \begin{align*} A=U\Gamma ...
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110 views

Numerical linear algebra (pseudoinverse of a matrix)

Let $A$ be the matrix: $$\left(\begin{matrix} \alpha I_{n} \\ \beta I_{n} \end{matrix}\right)$$ where $\alpha,\beta\in\Bbb C$ are not both zero. Derive (a) the (reduced) QR factorization of $A$ ...
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88 views

Iterative Method for a special kind of Sparse Matrix

I've the following problem.I've a sparse square Matrix $\bf M$. I can write $\bf M$ as: $${\mathbf M} = \begin{bmatrix}\mathbf A_{11} & \dots & \mathbf A_{1n} \\ \vdots & \ddots & ...
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191 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 ...