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

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64 views

Rayleigh quotient ($|r(q)-\lambda|=O(||q-x||_2^2)$ ?)

how to show $|r(q)-\lambda|=O(||q-x||_2^2)$ $r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, A is a Hermitian matrix. x is the unit eigenvector corresponding to $\lambda$. and q is a unit vector. ...
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1answer
76 views

Interpolation of polynomials

let $f(x)=2^x$ and $x_0=1$, $x_1=2$, $x_2=3$. Use divided differences to compute the interpolation polynomial $P(x)$ satisfying $P(x_i)=f(x_i)$, i=0,1,2 and $P'(x_1)=f'(x_1)$ and estimate error ...
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39 views

Is the following matrix Upper Hessenberg?

Does $$ A = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$$ properly satisfy the definition of upper Hessenberg?
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373 views

Applying Central Difference (Finite Difference Method) in MATLAB

I was given a rather complicated few problems to solve in MATLAB using the central difference method, and I'd like some help figuring out how to translate this into code. The goal is to discretize ...
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0answers
37 views

Show that Newton’s Method is well-defined for all k and converges to 0 for $x_0>0$

Let $f : R → R$ with $f$ twice continuously differentiable, $\gamma > f''(x)>\delta, f(0)=0,f'(x)>\rho $ for $x ≥ 0$. Show that for any $x_0 > 0$ that Newton’s Method is well-defined for ...
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1answer
29 views

Generalized Eigensystems

I am looking for solution algorithms for a second order generalization of the eigenvalue problem. A, B, and C are n-by-n matrices, I is the n-dimensional identity matrix, $\lambda_i$ is an unknown ...
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23 views

Jacobi Iteration with Shift

The question is to solve a linear system using Jacobi iterations with a shift of mu = 5. My code converges very quickly, but it does not yield the results that MATLAB gives with the backslash ...
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0answers
21 views

Convergence of recursive application of finite-difference operator to $C^{\infty}$ functions

Let $f\colon \mathbb{R}\to \mathbb{R}$ be an arbitrary smooth function (whose extension to a complex differentiable function is entire, if it matters). Let $\mathbf{D}_{h}$ be a finite difference ...
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0answers
113 views

Meaning of singular Jacobian and workarounds to Newton's method

I'm currently working with Galerkin's method to solve differential equations and I have to retrieve unknown coefficients for the truncate expansion. This is just to set the background for why I need ...
1
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1answer
48 views

Derivative of $\|Ax-b\|_1$

Using least squares approximation $E^2 = \| Ax - b\|^2 = (a_1x - b_1)^2+...+(a_mx-b_m)^2$ The derivative of E^2 at the point $\hat{x}$ is zero if: $(a_1\hat{x}-b_1)a_1+...+(a_m\hat{x}-b_m)a_m=0$ ...
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1answer
40 views

vector space of natural numbers

I wonder, is it possible for the natural numbers (with zero) t be a vector space on SOME field? I understand why it cannot be over real numbers because of muliplication with negative scalar. BUT what ...
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0answers
22 views

LU Decomposition for the solution of two linear systems

Let's say I have the following linear system: \begin{equation} \left[ \begin{array}{cccc} S&&L^{T}&&A^{T}&&0\\ L&&0&&0&&0\\ ...
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22 views

Applying perturbed matrix to unperturbed eigenvector

Suppose we've got a matrix $P$ and a perturbed version $\hat{P}=P+E.$ Given that $v$ is an eigenvector of $P$ with $Pv=0,$ I'd like to get as sharp a bound as possible on $\hat{P}v$ (in terms of ...
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1answer
34 views

convolution on 2 by 2 matrices

Let $m$ be a positive integer, and let $A_1,B_1 \in \operatorname{SL}(2,\mathbb{Z})$. Can one always find matrices $A_2,B_2 \in \operatorname{SL}(2,\mathbb{Z})$ such that $$ A_1 \left( ...
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0answers
213 views

Matlab project - Jacobi method for tridiagonal matrices…

I have to do a project in Matlab to my University and I don't quite understand what I should do. I was given script that solves systems of equations with Jacobi's method with given tolerance and ...
1
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2answers
50 views

A function to convert a vector to a number and vise versa?

Sorry in advance if I didn't choose the right tags for the question, I wasn't sure. So I'm a programmer and writing a saving/loading system for data. The way I was serializing (saving) vectors is via ...
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1answer
32 views

Enciphered Message with linear enciphering function.

My semester tests are coming up and as I was looking through past papers I came across this question. I was missing a lot during the beginning of the year and this was no doubt covered during my ...
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3answers
41 views

Divergence of fixed-point iteration for real starting values

Consider the linear system of equations $Ax = b$ with invertible $A\in \mathrm{GL}(n,\mathbb R)$ and $b\in\mathbb R^n$. For $A = M - N$ with invertible $M$ the solution $x_* = A^{-1}b$ is a fixed ...
2
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1answer
41 views

Weed out numerical artifacts from matrix inversion

I am working with the inverses to a set of large sparse matrices (in Matlab). A key indicator for my application is the number of non-zero entries in each row, and I recently discovered that I was ...
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0answers
73 views

Strictly diagonally dominant matrix -LU factorization

Let $A\in\mathbb{C^{n\times n}}$ be strictly diagonally dominant. I want to show that the LU factorizations with and without partial pivoting are the same for these matrices. For start, I created ...
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1answer
52 views

Why base vectors are 1 in length? [closed]

I can't really find any substantial reference in the math literature that justifies the fact that basis vectors are usually $\begin{bmatrix}1&0&0\end{bmatrix}$ or ...
2
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1answer
51 views

Is there a function in MATLAB that will estimate the initial condition from a set of data?

I was given a state-space model of a system and a list of outputs for t=0 to t=5, sampled every 0.1 seconds and asked to approximate the initial condition. Is there a function in MATLAB that will take ...
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0answers
28 views

Pairing Two Point Clouds

So I have two point clouds $X$ and $Y$ each with $N$ points in the familiar $\mathbb{R}^3$ euclidian 3D space. I then have an inter-point distance $d(\vec x_i,\vec y_j)$ which is zero if $\vec x_i$ is ...
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2answers
45 views

stuck with some parts of the proof about “ matrix is normal iff each of its eigenvectors is also an eigenvector of its transpose conjugate matrix”

When I read the book Iterative Methods for Sparse Linear Systems, Second Edition, I get stuck with the following proof. The yellow highlight parts are the positions I have trouble to understand. ...
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1answer
73 views

how to prove the relationship about spectral radius, numerical radius and matrix two norm?

When I read page 24 in Iterative Methods for Sparse Linear Systems, Second Edition, I can not understand the following statement: (My major is not math) Let $A$ be an n-square complex matrix with ...
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0answers
31 views

Relative Error with Respect to Frobenius Norm

I'm look at this tiny book called "Deblurring Images: Matrices, Spectra, and Filtering" by Hansen, Nagy, O'Leary. This is a self study, but I believe my question is broad enough so that it can be of ...
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20 views

Dimensionality Reduction

Let $X\in\mathbb{R}^{100\times 100}$ matrix and let its eigenvector and eigenvalues be $X_{vec}$ and $X_{val}$ respectively. If the rank of $X$ is $5$, then is it possible to approximate $X$ with ...
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3answers
79 views

Is $\bigl\|\frac{vv^T}{v^Tv}\bigr\|=1$? For any vector $v\in \mathbb{R}^{n}$

I am stuck while showing that $$\biggl\|\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $v\in \mathbb{R}^n$, and $\|.\|$ is a matrix norm. Here is my steps: I used Frobenius norm: A Frobenius matrix ...
0
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1answer
78 views

How does the Simplex method of solving LPs use the starting solution?

Say one looks at the LP (in slack form) and sees that assigning $0$s to all the non-basic variables doesn't give a valid solution but some other non-trivial assignment of values to the non-basic ...
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2answers
29 views

What does it mean by one matrix is **unitarily similar** to another?

I am reading a tutorial about the Lanczos method for eigen problem / SVD. It mentioned "Then the tridiagonal matrix $B^∗B$ is unitarily similar to $A^∗A$. " What does it mean? I can derive this: ...
1
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1answer
95 views

conjugate gradient method for semi definite case

Show for a symmetric, positiv semi definite matrix $A$, a vector $b\in Ran(A)$ and initial vector $x_0$: (1) All directions $d_0,d_1,...,d_m$ of the conjugate gradient method are in the range ...
0
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2answers
221 views

Definition of Spectral Radius / Eigenvalues of Product of a Matrix and its Complex Conjugate Transpose

For any matrix $\textbf{A}$, I know that in the Euclidean L2 induced/operator norm, $\|\textbf{A}\|=\sqrt{\rho(\textbf{A}^{*}\textbf{A})}$, with * being the complex conjugate transpose and $\rho$ ...
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0answers
54 views

Condition number perturbation

I have a matrix of the form $\tilde{H} = H + i A A^\dagger$. It is known that $H$ is hermitian and that $\tilde{H}$ is invertible and $A A^\dagger$ has a kernel of dimension $\geq 1$. I want to study ...
2
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1answer
35 views

Finding An Orthogonal Matrix

Let $u = (0,1,2,2)^{T}$, $v = (-3,0,0,0)^{T}$. Find an orthogonal matrix $A$ such that $Au=v$ and $A = I-B$, where $B$ is a matrix of rank one. I started by writing $A$ as $A = I - xy^{*}$ and using ...
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0answers
42 views

Computing a few eigenvalues of large sparse nonsymmetric matrix without LU factorisation

I'm trying to find a few targeted eigenvalues of a large sparse (N=1e6,nnz=4e6) non-symmetric matrix. Currently I'm using MATLAB's 'eigs' function with the 'sigma' option and this uses the Shifted ...
0
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1answer
40 views

Psuedo-inverse of block low-rank, symmetric matrix?

I have a matrix that looks like $$ D = \left[ \begin{matrix} c_1aa^T & c_2ab^T \\ c_2ba^T & c_3bb^T \end{matrix} \right] $$ where $c_1, c_2, c_3$ are scalars and $a, b$ ...
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0answers
44 views

the Gauss-Jordan algorithm requires how many multiplications/divisions and add/subtractions

I am trying to show this following result. The Gauss-Jordan algorithm requires $\frac{n^3}{2}+n^2-\frac{n}{2}$ multiplications/divisions and requires $\frac{n^3}{2}-\frac{n}{2}$ ...
2
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1answer
72 views

derivation of GMRES question: why is my result for the approximate solution to $Ax=b$ always exact?

I am trying to see if I understand the GMRES method and it's result. But somewhere I get confused and I wonder if I am making a mistake. We start with a system $Ax=b$. We look for approximate ...
0
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1answer
46 views

Rank of the evaluation of a polynomial matrix

Given a polynomial matrix $A(t)$ of rank $r$, I would like to know at what complex evaluations of $t$ the rank decreases. Some research with google told me these values are sometimes called the zeros ...
1
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1answer
59 views

Connection between results of two SVDs

Consider SVD of $M$: $$ M = U \Sigma V^\top $$ And SVD of $N= \ln M$: $$ N = U^\prime \Sigma^\prime V^{^\prime\top} $$ Anyone knows/has seen/can think of any interesting connection/relation between ...
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35 views

Recasting Ax=b to use SOR method

Just need some guidance as to how to recast the matrix equation equation $Ax = b$ so that I can produce an iterative matrix to perform Succesive Over Relaxation on. These matrices are $n x n$ I ...
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1answer
31 views

Is this a reasonable method of numerically comparing two matrix functions?

I am currently trying to compare two matrices with elements which are too complicated for me to algebraically show that they are equal element wise and I decided to try the following approach: ...
2
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0answers
36 views

Can the Lanczos algorithm converge very fast by taking a good initial guess?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
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0answers
17 views

Properties of specific case of the gradient descent method

Consider the gradient descent method for the system $Ax=b$ where, $A=\begin{pmatrix}1 & 0 \\0 & a\end{pmatrix}, b=\begin{pmatrix}0 \\ 0\end{pmatrix}$ and the initial vector ...
0
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1answer
37 views

Why outer product isn't backward stable

Outter product isn't backward stable. This is because output matrix most likely has rank one and thus can't be represented in the form $(x + \delta x)(y + \delta y)^*.$ Also we know that if output ...
0
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3answers
27 views

Matrix Norm Division

Suppose $A=uv^*$ where $u$ is an $m$-vector and $v$ is an $n$-vector. For any $n$- vector $x$, we can bound $||Ax||_2$ as follows: $||Ax||_2 = ||uv^*x||_2=||u||_2|v^*x|\leq||u_2||||v||_2||x||_2$. ...
0
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1answer
91 views

Solve Bratu problem using Python

I am going crazy trying to solve the Bratu problem using Python: $$y''(x)+ e^{y(x)} = 0, \quad \lambda = 1, \quad x \in(0,1),$$ $$y(0) = y(1) = 0$$ I have to solve this using the tridiagonal ...
0
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1answer
51 views

Uniqueness of Thin QR Factorization.

Let $A \in \mathbb C^{m x n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m x n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal elements, ...
2
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2answers
41 views

How to show that $\| QA\|_2=\| A \|_2$ where $Q$ is unitary (for a matrix A)

I want to show that for a unitary matrix $Q$ and a matrix $A$ that $$ \|QA\|_2=\|A\|_2$$ I start with the definition of matrix induced norms: $$\| QA \|_2 = \sup_{x \neq ...
0
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1answer
52 views

What is the computational cost of reduced row echelon and finding the null space?

I'm taking computational linear algebra, and haven't been able to find too much information about the computational cost (in terms of m=rows and n=cols) of these two routines: Reduced Row Echelon ...