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

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How can I know the stiffness

How can I formulate this system? I would like to know the stiffness of this system. $$\begin{cases} x'_1=-0.01x_1+10^6x_2x_3 \\ x'_2= 0.01x_1-10^6x_2x_3-5\cdot10^{7}x_2^2 \\ ...
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10 views

Does the Conjugate Gradient Method provide an eigenvalue estimate?

Suppose that we apply a Krylov subspace method to the linear system $A x = b$. For example, if $A$ is symmetric positive-definite, then the Conjugate Gradient method may be used. I remember that the ...
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1answer
26 views

Norm of Outer Product

Let $x \in \mathbb{R}^N$ and $ y\in \mathbb{R}^M$. Show that $\|xy^T\|_{\infty}=\|x\|_{\infty}\>\|y\|_1$ I've been able to show the following: $\|xy^T\|_{\infty}= \|xIy^T\|_{\infty} \le ...
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1answer
24 views

Is LU decomposition of matrices efficient for today's standards?

This is in the spirit of a previous question of mine about the efficiency of the QR algorithm. The reason for asking is that I want to motivate some students, and I'm also curious. I do understand ...
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29 views

Nearest points / residuals on a total least squares parabola

Consider fitting a parabola $y = a + bx + cx^2$ to 2d data $X_i, Y_i$ with noise in both X and Y, using the the singular value decomposition as in Total_least_squares (TLS): $\qquad X = [ 1\ \ Xdata\ ...
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16 views

Enstrom and kakeya theorem

i need some numerical application of enestrom Kakeya thereom ? in fench je cherche une application du theoreme d enstrom et kakeya application numerique ou une application dans un domaine de ...
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1answer
31 views

Heat equation in 1D with collocation method

I want to use the collocation method to solve $u_t=u_{xx}$. I impose the PDE pointwise and expand the solution in Fourier Series: $$ \partial_{t}\sum_{k=-K}^{K}\hat{u}_{k}(t)\ ...
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0answers
14 views

LU-factorisation of a square matrix

I need to show that the following matrix cannot be factor into the product LU. \begin{equation} A=\begin{bmatrix}1&2&-1\\2&4&0\\ 0&1&-1\end{bmatrix} \end{equation} I did the ...
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24 views

Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve $Ax = b$, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous ...
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15 views

Solving a structured partitioned linear system

I am trying to solve the following partitioned linear system, where each letter represents a block $\begin{pmatrix}-H & A^T & I_n \\ A & 0_1 & 0_2 \\ z_D & 0_2^T & ...
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22 views

Numerical analysis and localization of the roots of a polynomial [closed]

I want a digital Matlab application of root location of a polynomial theorem: Enestrom and Kakeya theorem.
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1answer
27 views

A problem about lub and glb of matrix

For any matrix $A\in \mathbb{C}^{n\times n}$, define $$lub_K(A):= \inf\{\alpha\geq 0: AK\subset \alpha K\},$$ and $$glb_K(A):= \sup\{\alpha\geq 0: \alpha K\subset AK\},$$ where $K$ is a equilibrated ...
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1answer
14 views

Linear independence of three simple functions (2.9-22)

Why is the following set of three functions linearly independent on an interval $I$ with $x>0$ if $k_2$ can take on any value besides zero and still hold true for a zero sum or that $k_1=k_3$ which ...
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1answer
57 views

subtraction between sum of all elements of two symmetric matrices

Let assume that I have an $n\times n$ symmetric matrix $A$ and I know $A^{-1}$. Now, I have a new matrix $$M = \begin{pmatrix} A & b \\ b^T & c \end{pmatrix},$$ where $b$ is a vector and ...
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1answer
26 views

How to prove or disprove the matrices formula

Could some one give me some hints about the following prove of disprove: (a) If $PXX^TP^T=QXX^TQ^T$, then $PX=QX$; (b) If $PXX^T=QXX^T$, then $PX=QX$. In the above formulas, $P$, $Q$ and $X$ are ...
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2answers
437 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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19 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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1answer
37 views

Eigenvectors of transition matrices in PageRank algorithm

In my probability course, we were discussing applications of Markov Chains to computer science -- in particular, how the PageRank algorithm goes about finding stationary distributions, and thus, ranks ...
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2answers
41 views

How to solve $Ax=b$ via backward and forward substitution on Matlab

How can I solve $Ax=b$ in Matlab code via LU factorization. I know that the command [L,U]=LU(A) stores the ...
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3answers
76 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 ...
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0answers
29 views

Wiedemann for solving sparse linear equation

I am new member. I am researching in Wiedemann algorithm to find solution $x$ of $$Ax=b$$ Firstly, I will show a Wiedemann's deterministic algorithm (Algorithm 2 in paper Compute $A^ib$ for ...
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1answer
26 views

help to find SOR optimal parameter $w$

Please let following linear system as $Ax=b$: $$\begin{array}{l} 6a{x_1} + {x_2} + {x_3} = 1\\ {x_1} - 3a{x_2} + 4{x_3} = 2\\ {x_1} + {x_2} - 2a{x_3} = 3 \end{array}$$ Help me to prove that the ...
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3answers
123 views

Range and kernel of a linear transformation are ALWAYS disjoint

Is it true that the Range and kernel of a linear transformation are ALWAYS disjoint. I think they are not but I remember in my notes that the ker L= Im (L') this was under projections. So I am unsure ...
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10 views

choice of iterative linear system method

while implementing an unconstrained optimization problem, using Newton's method, I am faced with a Hessian matrix that is very large (10^8 by 10^8) but very, very sparse - Non zero elements along the ...
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43 views

Induced matrix p-norm

Let $\|\cdot\|_p$ denote the $p$ norm $(p≥1)$ defined for every vector $x=(x_1,x_2,\ldots,x_n)^t\in\mathbb C^n$ by $\|x\|_p=(\sum|x_j|^p)^{1/p}$ and let $|||\cdot|||_p$ denote the matrix norm defined ...
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26 views

What's the point of 1-norm matrix estimation? Why not brute force?

Calculating (brute-force) 1-norm of a square matrix should take $O(n^2)$ operations, with a small factor involved. Apparently, there is an algorithm (link) for estimating 1-norm that takes $O(n^2 t)$ ...
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2answers
58 views

Calculating the determinant of $A$ with $A_{ij}=a$ for $i<j$, $A_{ij}=-a$ for $i>j$, $A_{ii}=x$, using a pen and paper

Let $$A = \left[\begin{array}{cccccc} x&a&a&a&\dotsm&a\\ -a&x&a&a&\dotsm&a\\ -a&-a&x&a&\dotsm&a\\ -a&-a&-a&x&\dotsm&a\\ ...
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1answer
493 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 ...
3
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2answers
58 views

Good Textbook in Numerical PDEs?

I am currently taking a course on Numerical PDE. The course covers the following topics listed below. Chapter 1: Solutions to Partial Dierential Equations: Chapter 2: Introduction to Finite ...
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1answer
21 views

Finding the minimum of Condition number for this matrix

Suppose $A=\left[ {\begin{array}{*{20}{c}} {0.1\alpha }&{0.1\alpha }\\ 1&{1.5} \end{array}} \right]$. How can we find minimum of condition number $k(A)=\Vert A\Vert \Vert A^{-1} \Vert$ (Assume ...
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0answers
31 views

Is it better to compute $A^tA$ once and then $Ax$ several times or compute $y=Ax$ and then $A^ty$ every time?

So I have this algorithm which given a matrix $A$ it assigns $A=A^tA$ outside the loop and then on the algorithm loop it solves multiple instances of $Ax$ for different $x$s, (meaning that it's ...
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1answer
293 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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1answer
492 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 ...
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1answer
128 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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2answers
525 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: ...
2
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1answer
36 views

Restoring matrix from covariance matrix

Given $ A^TA $, how to restore $A$? (Any $A$ which produces this $ A^TA $). Given matrix $ A^TA $, vector $b$, and vector $Ab$, how to restore a matrix $A$ ? Correction to 2: The vector $A^Tb$, ...
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1answer
47 views

Why won't my conjugate gradient algorithm work?

I made this simple Conjugate Algorithm on Matlab n = length(b); r0 = b - A*x0; p0=r0; k=1; n0=(r0')*r0; while n0 >= eps && k <= n ...
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0answers
33 views

update cholesky factorization

I need to compute cholesky(H'*H) where H is a big sparse rectangular matrix. After that H is modified by adding several lines. That is Hn = [H ; line_1 ; ... ; line_n] in Matlab. How can I recompute ...
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37 views

matrix with positive diagonal elements

I was wondering if a symmetric matrix with positive elements only in the diagonal (negative elsewhere) is any special beside the symmetry. Thanks in advance
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0answers
36 views

Find the solution of linear equation using Wiedemann/ Krylov method

I am using Wiedemann (some literature called Krylov method) to find the solution of a linear equation that defined as $$Mx=b$$ Instead of resolving entire elements of x (size $K \times 1$), we can ...
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0answers
53 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
18 views

If normal linear regression finds $A$ such that $AX \sim Y$, then how do I solve $BAX \sim Y$?

If normal linear regression finds $A$ such that $$AX \sim Y$$ then how should I solve $$BAX \sim Y$$ where $B$, $X$ and $Y$ are given (non-invertible) matrices? I could of course derive the solution ...
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1answer
182 views

How can I tell which matrix decomposition to use for OLS?

I want to find the least squares solution to $\boldsymbol{Ax}=\boldsymbol{b}$ where $\boldsymbol{A}$ is a highly sparse square matrix. I found two methods that look like they might lead me to a ...
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2answers
45 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
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18 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$_{k}$, ...
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18 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 ...
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1answer
30 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 ...
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0answers
26 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 ...
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1answer
16 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, ...
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1answer
19 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 ...