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

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4
votes
2answers
35 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
0
votes
0answers
14 views

order of convergence for approximations

Let $u \in L^{2}(0,1)$ and $0 < x_{1}< x_{2}<... < x_{n} = 1$, where x$_{k}$ = k$\cdot$h, n$\cdot$h = 1, a partition of the interval [0,1]. Define I$_{k}$(x) = 1 if x $\in$ [x$_{k}$, ...
0
votes
0answers
11 views

Location and perturbation of eigenvalues

This is a problem from Horn and Johnson's Matrix Analysis. I'm having trouble showing the bolded parts in the following paragraphs. In fact, I don't really understand what the sentences mean. I would ...
0
votes
1answer
25 views

Eigenvalue inequalities for Hermitian matrices

This is a problem from Horn and Johnson's Matrix Analysis. I've tried to follow the problem but I can't find a way to lead to the conclusion the problem is suggesting. Any solutions, hints, or ...
0
votes
0answers
21 views

A problem on Gersgorin cirle passing through the eigenvalue of an absolute matrix

I'm having trouble solving the following problem. I think I need to show that the matrix $D^{-1}|A|D$ has property SC, but I can't come up with a way to show it. I would really appreciate any ...
0
votes
1answer
14 views

A problem about a theorem on irreducible matrix

I'm stuck on a problem where I need to find a counterexample. I'm not sure how to come up with a reducible matrix to show that it doesn't satisfy the result of the following corollary. Any solutions, ...
0
votes
1answer
16 views

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$ $\lt C_k'$, for at least one value of $k$

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$$\gt C_k'$, for at least one value of $k=1,\dots, n$, where $C_k'$ denotes $A$'s deleted absolute column sums ($a_{kk}$ is ...
2
votes
1answer
18 views

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. If $A$ is real, show that every eigenvalue of $A$ is real.

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. (a) If $A$ is real, show that every eigenvalue of $A$ is real. (b) If $A \in M_n$ has real main diagonal entries and its ...
2
votes
1answer
15 views

Show that the intersection taken over the Gersgorin discs of all similar matrices of $A$ $=$ $\sigma (A)$

Show that $\bigcap_S G(S^{-1}AS)$ $=$ $\sigma (A)$; the intersection is taken over all nonsingular $S$, and $\sigma (A)$ is the spectrum of $A$. I'm lost as how to even begin to prove this fact. Any ...
1
vote
0answers
23 views

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

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

Domain for which this matrix is positive definite

What is the domain for which this matrix is positive definite? $$\left(\begin{array}{cc} 12x^2 & 1 \\ 1 & 2 \\ \end{array}\right)$$ I'm trying to figure this out. I know the ...
0
votes
1answer
50 views

Quality of approximation [on hold]

Basically i have written a code in matlab that is used to solve lower triangular system with forward elimination Ax=B. However, the forward elimination script will not give the exact vector but a ...
0
votes
1answer
60 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 ...
0
votes
1answer
15 views

$\mathbb{R}^3$ 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 $\mathbb{R}^3$ with coordinates $(P_X,P_Y,P_Z)$ a plane Defined by: a point '$O$' ...
0
votes
2answers
404 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 ...
0
votes
1answer
18 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 ...
0
votes
0answers
29 views

Is the following matrix Upper Hessenberg?

Does $$ A = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$$ properly satisfy the definition of upper Hessenberg?
0
votes
0answers
16 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 ...
0
votes
0answers
32 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 ...
0
votes
0answers
30 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 ...
0
votes
0answers
8 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 ...
1
vote
0answers
24 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
vote
1answer
38 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: $(a_1\hat{x}-b_1)a_1+...+(a_m\hat{x}-b_m)a_m=0$ ...
0
votes
1answer
77 views

relative error relation

Let $x$ be a non-null quantity. Let $\hat{x}$ be its approximation. I am trying to find the relation between: $\frac{\left | x-\hat{x} \right |}{\left | x \right |}$ and $ \frac{\left | x-\hat{x} ...
0
votes
1answer
423 views

computing Moore-Penrose pseudoinverse when SVD computation does not converge

I am writing a routine to return the Moore-Penrose inverse of a rectangular matrix. Currently am computing the Moore-Penrose inverse using SVD, i.e., if the SVD is given by $A = \sum_{i=1}^r ...
2
votes
0answers
466 views

What is the Moore-Penrose pseudoinverse for scaled linear regression?

The matrix equation for linear regression is: $$ \vec{y} = X\vec{\beta}+\vec{\epsilon} $$ The Least Square Error solution of this forms the normal equations: $$ ({\bf{X}}^T \bf{X}) \vec{\beta}= ...
0
votes
1answer
119 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 $A$ ...
0
votes
1answer
31 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 ...
0
votes
0answers
6 views

Get stuck with some statements of convergence rate of the iteration method from “Iterative methods for sparse linear systems (2nd edition) ”

Here are the statements I get from the book and the two highlight parts are what I can not understand well. The questions are: Why we can conclude that $\rho=\rho(G)$ from"the above analysis"? ...
0
votes
0answers
20 views

Interpolation question about polinomials?

Let $f(x)=x^n$. show that for each $n$ distinct point $x_0,... x_n$ we have $f[x_0,...x_n]=0$ and $f[x_0,...,x_{n-1}]=\sum_{i=0}^{n-1}x_i$. also show that if $x_{i+1}-x_i$ is fixed for each $i\ge 0$ ...
1
vote
0answers
17 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\\ ...
0
votes
0answers
10 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 ...
3
votes
1answer
461 views

Trace Minimization of Covariance Matrix

Given a matrix X whose rows contain observations collected at some locations. Can someone explain how trace minimization of covariance matrix $XX^T$ can lead to orthogonal / mutually independent ...
0
votes
1answer
23 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( ...
0
votes
0answers
31 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 ...
0
votes
1answer
282 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 ...
0
votes
0answers
16 views

$Tn(x) =\frac{(x-\sqrt{x^2-1})^n+(x+\sqrt{x^2-1})^n}{2}$, show that $|Tn(x)| \le 1,\forall x \in [-1,1]$

$Tn(x) =\frac{(x-\sqrt{x^2-1})^n+(x+\sqrt{x^2-1})^n}{2}$, show that $|Tn(x)| \le 1,\forall x \in [-1,1]$
0
votes
0answers
11 views

Consistently obtaining the negative of the correct eigenvalue in hand calculations

The question is "find the eigenvector of A corresponding to the given eigenvalue" A = | 1 0 2 | |-1 1 1 | | 2 0 1 | Eigenvalue = -1 Add -1 * I to A Row reduced to | 1 0 1 | | 0 1 1 ...
1
vote
2answers
33 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 ...
1
vote
2answers
468 views

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: ...
1
vote
1answer
476 views

How to find the Householder transformation?

Assume $x=(1,0,4,6,3,4)^T$. Find a Householder transformation and a positive number $\alpha$ such that $Hx=(1,\alpha,4,6,0,0)^T$. I'm sorry that I don't know how to start with this problem. A ...
0
votes
0answers
30 views

QR Algorithm without Shifts (Trefethen and Bau)

A real symmetric matrix $A$ has eigenvalue 1 of multiplicity 8, while all the rest of the eigenvalues are $\leq 0.1$ in absolute value. Describe an algorithm for finding an orthonormal basis of the ...
0
votes
1answer
22 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 ...
4
votes
3answers
29 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 ...
0
votes
0answers
17 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 ...
2
votes
1answer
36 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 ...
2
votes
2answers
44 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 ...
0
votes
1answer
47 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 ...
1
vote
2answers
37 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. ...
0
votes
0answers
11 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 ...