Tagged Questions
1
vote
0answers
22 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
395 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 $n\in\mathbb{N}$
$$
...
8
votes
0answers
192 views
Limit of sequence of growing matrices
Let
$$
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),
$$
...
2
votes
1answer
225 views
Nonlinear equations and unique solution
How to show that the following system of equations has a unique solution $(x,y)$? $x,y$ are scalars.
$x+\frac{3}{4}y+\frac{1}{20}\sin x=0$
$-\frac{37}{40}x+y+\frac{1}{10}\sin y=0$
I tried ...
1
vote
1answer
212 views
The coefficients in the inverse of unit lower triangular Toeplitz matrix
I have a question about the inverse of lower triangular Toeplitz matrix.
There is a matrix $Q$ which can be infinite-dimensional.
$$
Q=\left[\begin{array}{ccccc}
1\\
-k_{1} & 1\\
-k_{2} & ...
0
votes
0answers
84 views
Resolve Ax=b using preconditionner for the conjugate gradient
I would like to resolve the equation $$
\left[
\begin{array}{ c c }
A & B\\
B' & 0
\end{array} \right]
\left[
\begin{array}{ c}
x \\
y
\end{array} \right] =
...
0
votes
0answers
130 views
Diagonal Dominance and Spectral Radius
For positive semi-definite matrices, $A$ and $B$ with real entries,
Let:
$X=I-(2Diag(A)-B)^{-1}(A-B)$
The spectral radius $\rho(X) \leq ||X||$.
As, $(2 Diag(A)-B)$ becomes a better approximation ...
1
vote
1answer
42 views
Vanishing ratio of norms implies vanishing ratio of individual elements?
Consider two vectors $x,y \in \mathbb{R}^n$ be parameterized by a value $t>0$, and suppose that
$$\lim_{t \rightarrow 0} \frac{|x(t)|}{|y(t)|}=0,$$
where $|\cdot|$ denotes the standard Euclidean ...
2
votes
1answer
43 views
Asymptotic convergence
I have a previous similar question. I'm working out that one with the answerer, but I'm trying to gain insight from a different angle, especially in approaching these problems.
I must establish that ...
2
votes
1answer
134 views
A question on norm of error vector
Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
2
votes
3answers
107 views
Convergence of a pair linearly independent elements of a vector space
Let $v_1$ and $v_2$ be linearly independent elements of some normed vector space $V$. We have a sequence $(a_n,b_n)$ and are told that $a_nv_1+b_nv_2$ converges to the zero element. Is it the case ...
2
votes
1answer
557 views
matrix multiplication convergence problem
Suppose A is a square matrix.
Does $A^n$(matrix multiplication) converge when n is an infinite big number？
Is it always true or under certain circumstances?
2
votes
1answer
53 views
solving coupled discrete systems
Suppose you have a discrete system, whose evolution is governed by the following equations:
$\mathbf{x}[k+1] = f_1(\mathbf{F}[k], \mathbf{x}[k])$
$\mathbf{F}[k+1] = f_2(\mathbf{F}[k], ...
0
votes
1answer
143 views
convergence rate of matrix product
Suppose you have a linear system like this:
$$\mathbf{x}[k+1] = \mathbf{D} \mathbf{x}[k]$$
where matrix $\mathbf{D}$ is diagonal. Assume its diagonal entries are real, greater than zero and less than ...
0
votes
2answers
609 views
Looking for a proof for the convergence of matrix geometric series
Consider a symmetric matrix $A$ with non-negative integer coefficients. It appears that the geometric series $\sum_{i \geq 0}A^i$ will converge to a matrix $B$ if the spectral radius (the largest ...
0
votes
1answer
129 views
does the following dynamic system converge to a steady state?
This is an economics problem, but I'm pretty sure this kind of thing comes up elsewhere. I've used dynamic programming to find the optimal path of a system (law of motion), which is:
...
3
votes
2answers
410 views
Does $M_n^{-1}$ converge for a series of growing matrices $M_n$?
$M_n$ is a $n\times n$ matrix with $M_{n+1}=\begin{pmatrix}M_n & a_n \\ b_n^T & c_n\end{pmatrix}$ and $a_n, b_n, c_n \to 0$ for $n\to \infty$. Is this sufficient to state $$ ...
3
votes
3answers
285 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?
