Tagged Questions

21 views

Euclidean norm preserving linear transformation: name and characterization?

If $T$ is a linear transformation in $\mathbb{R}^{n}$ (from itself to itself) that preserve Euclidean norm, then for any $\vec{v}$ then $(T\vec{v})\cdot(T\vec{v})=\vec{v}\cdot\vec{v}$. We already know ...
18 views

52 views

63 views

Polynomial Interpolation

My professor gave the following question as a practice for study guide. Any assistance in terms of helping me to solve this would be much appreciated. Suppose that $f$ is continuous and has ...
130 views

cube root of positive definite matrix

Suppose that $A$ is a real symmetric positive definite $20\times 20$ matrix with condition number $\kappa\le 1000$. I want to solve the system of linear equations $$A^{1/3}x=b$$ with $10$-digit ...
37 views

Algebra question / conversion of ranges

Greets All Forgive me if I'm using the wrong terms but I'm trying to sync up two number ranges together. Example: I have two x axis (ranges) I would like to equate with each other ...
47 views

Fredholm integral equation of first kind

I want to solve the Fredholm integral equation of first kind: $$\int_L K(x,y)U(y)dy = f(x)$$ in these equation the function $U(y)$ is the unknown and the so-called kernel $K$ and the right hand side ...
78 views

Solving Poisson Equation Finite-difference using maple

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

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, ... 1answer 63 views power of a matrix+induction I am trying to solve the following problem: Let A the block form A= \begin{bmatrix} B &C \\ 0&I \end{bmatrix} in which the blocks are n \times n. Prove that if B-I is ... 0answers 22 views Nontrivial Matrix-estimate I try to proof the following estimate: \begin{align} h' W^{-1} H W^{-1} h \geq c h' H h \qquad c>0, \qquad\qquad (1) \end{align} where h\in\mathbb{R}^{K-1} and ... 1answer 31 views FETI domain decomposition - kernel of local stiffness matrices Consider the differential equation \begin{align*} -\Delta u&=f\mathrm{\ in\ }\Omega \\ u & = 0 \mathrm{\ on\ }\partial\Omega \end{align*} with \Omega=(0,1)^2. We're splitting \Omega into ... 1answer 20 views l_1 Matrix Norm Inequality I am independently studying Numerical Analysis and came upon the following question: l_1 vector norm ||x_1|| is defined as ||x_1||=\sum|x_i|. How can we show that for the natural matrix ... 0answers 31 views overdetermined system I'm trying to solve an overdetermined system with Second-order differential equations. Without noise everything works fine. When I add Gaussian noise the solution is not stable anymore and I've got a ... 0answers 23 views Difference Equations and displacement operator For a Prep exam Exercise from the book: Numerical analysis of scientific computing. Section 1.3-3 Let p be a polynomial of degree m, with p(0) \neq 0. If a sequence x contains m ... 0answers 55 views Inverse of Sum of Matrix Inverses Given N positive-definite matrices \Lambda_i, I need to efficiently compute \Gamma_N, where \Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}. $$Applying the Woodbury matrix identity ... 0answers 50 views This system is contractive? I have a system which has a form of find point problem, described as following$$p_i=h_i(\mathbf{p})$$where$$p_i\in[0,1]$$is the i-th components of the n-dimensional column vector ... 1answer 28 views A=N-P decomposition convergence to true solution I'm taking a numerical analysis course and we're looking at an iterative method for solving Ax=b where A is a square matrix, with a A=N-P decomposition with the formula$$Nx^{k+1}=Px^k+b$$... 1answer 71 views Finding a Unit Vector v for a Matrix A such that the 2-norm of AV is equal to the 2-norm of A I have been working on the following problem: Let A be the following 2x2 matrix: A = [1 1; 0 1] (MATLAB notation) Find the 2-norm of A and a unit vector v such that the 2-norm of Av = the 2-norm of ... 1answer 113 views What is the upper bound on the error of a matrix multiplication When both A and B are n x n upper-triangular matrices, the entries of C = AB are defined as follows:$$ c_{ij} = \begin{cases} \sum _{k=i}^ja_{ik}b_{kj} & 1\leq i\leq j\leq n \\0 & 1\leq j\lt ...
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 ...