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

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1answer
25 views

Eigen value system? solution

I have the following system. $AW = \lambda B W$ Where $A,B,W$ are matrices and $\lambda$ is a scalar. The values of $A,B$ and $\lambda$ are known. $B$ is invertible. This is a solution to an ...
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0answers
12 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 ...
1
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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 ...
0
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1answer
29 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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0answers
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 ...
1
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0answers
15 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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0answers
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 ...
2
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0answers
16 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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1answer
32 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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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 ...
1
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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 ...
0
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1answer
60 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 ...
1
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1answer
27 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 ...
2
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1answer
38 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 ...
2
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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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0answers
11 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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0answers
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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0answers
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 $O(n^2 t)$ ...
3
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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\\ ...
0
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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 ...
3
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2answers
60 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 ...
2
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1answer
37 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 $A^Tb$, ...
0
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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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0answers
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
2
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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 ...
1
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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 ...
0
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0answers
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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0answers
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 ...
0
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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 ...
0
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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, ...
0
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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 ...
2
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1answer
19 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
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1answer
26 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
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2answers
30 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 ...
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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. ...
0
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1answer
69 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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0answers
30 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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0answers
55 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
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0answers
33 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
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1answer
20 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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0answers
20 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
12 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
33 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
42 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
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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 ...