1
vote
1answer
53 views

Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks. ...
2
votes
1answer
43 views

Existence of solution for matrix equation $ (I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
1
vote
0answers
66 views

Is this a geometric series?

A geometric series is, in general, defined by: $$ \sum_{k=0}^{n-1}a\cdot r^{k}=a\cdot\dfrac{1-r^n}{1-r},\quad\quad \quad\quad \quad\quad r\neq1 $$ If I have instead the following: $$ ...
2
votes
1answer
43 views

Condition for convergence

Let $A \in \mathbb{R^{m\times{n}}}$ with full row rank. Let $B=I-\lambda A^T(AA^T)^{-1}A$ with $\lambda \in \mathbb{R}$. Determine the set of values of $\lambda$ for which $\exists \lim_{k \to ...
1
vote
0answers
24 views

Matrix vector multiplication convergence

Suppose we have a $n \times n$ matrix $M$ and a $n$-dimensional vector $s_0$. And $s_{k+1}=M*s_k$. What properties does $M$ have to have for that iteration to converge?
0
votes
0answers
17 views

Non-monotonic Gauss-Seidel components

I have noticed that on a 2D positive definite linear problem, after the first iterations, each of the iterate components converge monotonically to their stationary value. When viewing the GS method as ...
1
vote
1answer
53 views

Rate of convergence in an infinite geometric series of matrices

I have the following system $Z=[A^t + A^{t-1}+A^{t-2}+....+I]*E$, in which $A$ is a $n\times n$ matrix and $Z$ and $E$ are $n\times1$ vectors. The eigenvalues of $A$ are all smaller than one and the ...
1
vote
0answers
71 views

Sequences of 'Rayleigh-like quotients' and their minima for a symmetric positive semi-definite matrix

Let $A$ be an $N\times N$ symmetric positive semi-definite matrix with eigenvalues $0 \leq \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_N$ and corresponding eigenvectors $u_1, u_2, \dots, u_N$. ...
0
votes
1answer
52 views

Can this matrix really be used as a preconditionner?

I've read Boxerman's thesis and I feel that there is possibly a mistake. We have to resolve $$Ax=b$$ $A$ is a positive-definite symmetric matrix and is very sparse so the conjugate gradient method ...
1
vote
0answers
29 views

Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?

Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow, ...
15
votes
3answers
428 views

A question on convergence of series

Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$ $$ ...
21
votes
0answers
836 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
3
votes
1answer
523 views

Nonlinear equations and unique solution

How to show that the following system of equations has a unique solution $(x,y)$? $x,y$ are scalars. $x+\frac{3}{4}y+\frac{1}{20}\sin x=0$ $-\frac{37}{40}x+y+\frac{1}{10}\sin y=0$ I tried ...
1
vote
1answer
457 views

The coefficients in the inverse of unit lower triangular Toeplitz matrix

I have a question about the inverse of lower triangular Toeplitz matrix. There is a matrix $Q$ which can be infinite-dimensional. $$ Q=\left[\begin{array}{ccccc} 1\\ -k_{1} & 1\\ -k_{2} & ...
0
votes
0answers
110 views

Resolve Ax=b using preconditionner for the conjugate gradient

I would like to resolve the equation $$ \left[ \begin{array}{ c c } A & B\\ B' & 0 \end{array} \right] \left[ \begin{array}{ c} x \\ y \end{array} \right] = ...
1
vote
0answers
249 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
1
vote
1answer
44 views

Vanishing ratio of norms implies vanishing ratio of individual elements?

Consider two vectors $x,y \in \mathbb{R}^n$ be parameterized by a value $t>0$, and suppose that $$\lim_{t \rightarrow 0} \frac{|x(t)|}{|y(t)|}=0,$$ where $|\cdot|$ denotes the standard Euclidean ...
2
votes
1answer
48 views

Asymptotic convergence

I have a previous similar question. I'm working out that one with the answerer, but I'm trying to gain insight from a different angle, especially in approaching these problems. I must establish that ...
2
votes
1answer
145 views

A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
2
votes
3answers
116 views

Convergence of a pair linearly independent elements of a vector space

Let $v_1$ and $v_2$ be linearly independent elements of some normed vector space $V$. We have a sequence $(a_n,b_n)$ and are told that $a_nv_1+b_nv_2$ converges to the zero element. Is it the case ...
2
votes
1answer
1k views

matrix multiplication convergence problem

Suppose A is a square matrix. Does $A^n$(matrix multiplication) converge when n is an infinite big number´╝č Is it always true or under certain circumstances?
2
votes
1answer
69 views

solving coupled discrete systems

Suppose you have a discrete system, whose evolution is governed by the following equations: $\mathbf{x}[k+1] = f_1(\mathbf{F}[k], \mathbf{x}[k])$ $\mathbf{F}[k+1] = f_2(\mathbf{F}[k], ...
0
votes
1answer
161 views

convergence rate of matrix product

Suppose you have a linear system like this: $$\mathbf{x}[k+1] = \mathbf{D} \mathbf{x}[k]$$ where matrix $\mathbf{D}$ is diagonal. Assume its diagonal entries are real, greater than zero and less than ...
0
votes
2answers
868 views

Looking for a proof for the convergence of matrix geometric series

Consider a symmetric matrix $A$ with non-negative integer coefficients. It appears that the geometric series $\sum_{i \geq 0}A^i$ will converge to a matrix $B$ if the spectral radius (the largest ...
0
votes
1answer
202 views

does the following dynamic system converge to a steady state?

This is an economics problem, but I'm pretty sure this kind of thing comes up elsewhere. I've used dynamic programming to find the optimal path of a system (law of motion), which is: ...
4
votes
2answers
436 views

Does $M_n^{-1}$ converge for a series of growing matrices $M_n$?

$M_n$ is a $n\times n$ matrix with $M_{n+1}=\begin{pmatrix}M_n & a_n \\ b_n^T & c_n\end{pmatrix}$ and $a_n, b_n, c_n \to 0$ for $n\to \infty$. Is this sufficient to state $$ ...
3
votes
3answers
325 views

Does the ratio of consecutive terms converge for all linear recursions?

Does $f(n+1)/f(n)$ converge as $n\rightarrow\infty$ for $f(n)$ defined by a linear recursion, for all linear recursions?