Tagged Questions

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0answers
41 views

Approximation of (FEM) by (FDM)

[Ciarlet 3.4-6] Consider the functional $$J_h : v = (v_i)\in\mathbb{R}^N\longrightarrow J_h(v)\ :=\ \frac{h}{2}\sum_{i=1}^N\left(\left[\frac{v_{i+1}-v_i}{h}\right]^2 + c_iv_i^2\right) - ...
3
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1answer
36 views

Polynomial Condition Number

I have a question, from "Applied Numerical Linear Algebra"(James W. Demmel), that I don't know how to do. Consider $\mathbb{R^{d+1}}$ as the set of polynomials of degree $\leq d$ and $S_a$ the set of ...
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1answer
35 views

Hessenberg Matrices

$$A=\begin{bmatrix} 2&3&4\\ 3&-5&5\\ 4&5&0\end{bmatrix}$$ Find a unitary matrix $Q$ such that $A = QHQ^{H}$, where $H$ is Hessenberg. I am having a little trouble ...
3
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1answer
26 views

Arrange the computation of $f(x)$ so that the loss of significant figures can be reduced using $4$-digit decimal arithmetic

For the function $f(x)=\sqrt{x^2+1}-x$, have computed $f(100)=0$ using 4-digit decimal arithmetic (rounding after every intermediate calculation). The value $f(100) = 0.0049998750$ (to 8sf) is given ...
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0answers
32 views

commuting a LU factorisation

Consider the permutation matrix $P= \begin{pmatrix} 0 & \cdots & 0 & 1 \\ 0 & \cdots & 1 &0 \\ \vdots & \ddots & \vdots & \vdots \\ 1 & \cdots ...
2
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0answers
50 views

QR decomposition help

What do Q and R stand for? Why must the diagonal entries of R be positive instead of just nonzero?
1
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1answer
88 views

Proof of GMRES convergence

I am working on a homework, where we have to proof that GMRES finds an exact solution of $Ax = b$ in at most m steps (with A being an $m \times m$-matrix). The proof is split up into several steps, ...
2
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1answer
279 views

Matlab Matrix Multiplication Calculate Significant Figures

First off, long time reader, first time poster. Thanks in advanced for all the help this site has offered! So the question! I have two matrices in the form of the variables ...
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1answer
140 views

SVD and linear least squares problem

Edit: I've actually found an error: Instead of full SVD I had to use, "economy size" SVD, where $U$ has only first $n$ columns, and $\Sigma$ becomes a square matrix. I also forgot to take the ...
1
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1answer
52 views

Eigenvalues, Eigenvectors and Eigendecomposition

If there is a symmetric matrix, say $$B = \left[\begin{array}{cc} 0 & A\\ A^T & 0 \end{array}\right]$$ where $A$ is a $m\times n$ submatrix with $m \geq n$. Is it possible to express the ...
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2answers
64 views

solving a matrix equation $X-I=a \cdot (X\cdot U^T + U \cdot X)$

I am trying to solve the $n \times n$ diagonal matrix $X$ in the following equation: $$X-I=a \cdot (X\cdot U^T + U \cdot X)$$ where $0<a<1$ is a given scalar, $U$ is a $n \times n$ given ...
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0answers
42 views

Deduce that every m x n matrix A with integral elements is equivalent to a matrix S in Smith normal form.

Show that if $D$ is a diagonal matrix with integral elements then there is a diagonal matrix $S$ in Smith normal form such that $D\sim S$. Deduce that every $m \times n$ matrix $A$ with integral ...
3
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1answer
94 views

Show the space spanned is an invariant subspace

Let $A$ be real and let $\lambda = \alpha + i \beta$ be a complex eigenvalue of $A$ with eigenvector $x + iy$, show that the space spanned by $x$ and $y$ is an invariant subspace of $A$. What I ...
1
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1answer
106 views

Proving the SVD Theorem by induction on the rank of $A$

This is an exercise and it is divided into steps. The first step says: Suppose $A\in\mathbb{R}^{m\times n}$ has rank 1. Let $u_1\in\mathbb{R}^n$ be a vector in $R(A)$ such that $\left \| u_1 \right ...
0
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1answer
78 views

Generating Gauss-Seidel hard system

I am writing various Gauss-Seidel algorithm parallel implementations using different programming techniques for my assignment. I have created a MATLAB script for generating strictly diagonally ...
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2answers
230 views

Given orthogonal basis what is the orthogonal complement?

The question states: Let $q_1,q_2,\dots,q_n$ be an orthogonal basis of $\Bbb R^n$ and let $S = \operatorname{span}\{q_1,q_2,\dots,q_k\}$, where $1 \le k \le n-1$. Show that $S^\perp = ...
3
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2answers
84 views

How can I prove $\mathrm{maxmag}(A)=\frac{1}{\mathrm{minmag}(A^{-1})}$, and $\mathrm{maxmag}(A^{-1})=\frac{1}{\mathrm{minmag}(A)}$?

Using $\mathrm{maxmag}(A)=\max_{x\neq 0}\frac{\|Ax\|}{\|x\|}$, and $\mathrm{minmag}(A)=\min_{x\neq 0}\frac{\|Ax\|}{\|x\|}$ I found this quite simple to prove using a proposition stating that ...
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2answers
55 views

Calculate sum of ONB

Here's a homework question: Let ${u_1, \ldots, u_n}$ be an ONB in $C^n$. Assuming that $n$ is even, compute $$||u_1 - u_2 + u_3 -\cdots - u_n||$$ I have no idea how to solve this. Can anyone help? ...
0
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1answer
197 views

Norm $\|A\|$ is not induced by any vector norm [duplicate]

Possible Duplicate: Subordinate matrix norm I have a question in my homework for Numerical Linear Algebra, which is as follows: Show that the norm $\|A\| = \max \limits_{i, j} |a_{i,j}|$ ...
3
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2answers
122 views

Fast inversion of a triangular matrix

I need to inverse a matrix $A$ given its $QR$ decomposition. It's a numerical task. It is told that the inversion should be "possibly cheap". But it does not look like I can do something more ...
1
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1answer
148 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 ...