0
votes
0answers
41 views

Conversion of continuous, linear stochastic system to discrete, LQR/LQG

I have the standard stochastic, linear time varying system $dx(t) = (A(t)x(t) + B(t)u(t))dt + G(t)dw(t) $ with $x(t_0) = x_0$ with quadratic cost $J = x(t_F)^TQ_Fx(t_F) + \int_{t_0}^{t_F}\left( ...
1
vote
2answers
63 views

Response of stationary distribution to perturbation of a stochastic matrix

Suppose I have a (left) stochastic matrix $P$, i.e. a non-negative matrix with column sums equal to 1. Its maximum eigenvalue will be equal to 1, and the corresponding eigenvector $\mathbf q$, if it's ...
0
votes
1answer
34 views

Regular matrix and regular stochastic matrix

We know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular if at a certain power all elements are positive. Question is how can I make the link between the ...
0
votes
5answers
157 views

where can I take an online Second course in Linear Algebra or second abstract algebra?

Hi i am looking for an online/independent study Second course in Linear Algebra or abstract algebra I and II? Can you point me in the right direction. I need college credit. I need help ASAP Even a ...
5
votes
0answers
144 views

Generating a stochastic matrix with a given second dominant eigenvalue

I need a procedure (iterative or otherwise) that, given a positive integer $N$ and a (possibly complex) number $\lambda$ such that $0 < \vert \lambda \vert < 1$, will be able to generate an $N ...
0
votes
1answer
140 views

Linear Algebra Stochastic Matrix and Markov Chains

I have a few true and false questions I need help with. Can someone please check my work? The product of two stochastic matrices is a stochastic matrix. This is false I found a counterexample. 2 ...
1
vote
1answer
50 views

Positive definitness of infinite dimensional matrices

Assume $M=(b_{i,j})_{ i,j=1}^{\infty}$ is an infinite dimensional matrix such that $b_{i,i}>0$ and $b_{i,j}=b_{j,i}$ for all $i,j\in\mathbb{N}$ (i.e., $M$ is symmetric with positive diagonal ...
0
votes
0answers
50 views

Absorption probability in 1D RW with asymmetric step sizes, $ x<0 $

What is the probability of absorption at $ 0 $, as a function of position $ x $, for a 1D random walk (on $ \mathbb{Z} $) with asymmetric step sizes? For example, suppose that you can take two steps ...
-1
votes
1answer
65 views

Proof that the square of a stochastic matrix is stochastic

We know that the square of a stochastic matrix is also stochastic, because the two-step transition matrix of a Markov chain is necessarily stochastic. However, in there another way to independently ...
0
votes
1answer
34 views

How to find $X_i$ from this equation

Suppose $X_i=\nu_i+\frac{m-i}{m}X_{i+1}+\frac{i}{m}X_{i-1},\quad 1\le i\le m$ where $X_0=X_{m+1}=0$. I need to find an expression for $X_i$ in terms of $v_i$, $i$, and $m$. I know how to find it ...
0
votes
0answers
57 views

Gaussian process

I've some truble with my homework. A Gausian process $(X_t,t\in T)$ is is a process whose finite-dimensional distributions are (generalized) Gaussian, i.e. $(X_{t_1},\ldots,X_{t_n})\sim ...
0
votes
2answers
78 views

Discrete Markov Transition Matrix

Well, I've been reading over the internet but I've been unable to find a straight answer. I've got a transition matrix for a Markov Discrete Chain. I've made the graph and according to my knowledge, ...
3
votes
2answers
174 views

Eigenvalues of a tridiagonal stochastic matrix

I've tried to calculate the eigenvalues of this tridiagonal (stochastic) matrix of dimension $n \times n$, but I had some problems to find an explicit form. I only know that 1 is the largest ...
0
votes
0answers
126 views

Variance of the first return time of a simple random walk on an hypercube graph

I am trying to solve this problem.... I have a simple random walk on a $d$-cube (finite graph). At each vertex of the graph, the particle chooses one of $d$ edges equally likely. I need to calculate ...
0
votes
1answer
129 views

Calculate the determinant of the matrix ad hence prove

By computing the determinant of $\lambda I-L$ where $L$ is the Leslie matrix, derive the Euler Lotka equation. $$L= \begin{bmatrix} b_{1} & b_{2} & \ldots & b_{w-1} & b_{w}\\ s_{1} ...
2
votes
1answer
65 views

Linear system with probabilities (algebra)

I have a small problem that delays my project , it seems I am stuck here loads of time. It is probably very easy but I cant see it right now , I am very anxious about this, please take a look: ...
7
votes
3answers
228 views

Singular covariance matrix

I am looking into the process $\{X_t, t\in\mathbb{Z}\}$, $X_t=A\cos(\lambda t)+B\sin(\lambda t)$, here $\lambda\in(0,\pi)$ is fixed, $A$ and $B$ are uncorrelated random variables with $EA=EB=0$, ...
2
votes
2answers
246 views

Looking for an example of a Markov Chain

I am looking for an example of a Markov Chain characterized by, say, 3 by 3 matrix that has more than one eigenvector (say a population distribution of birds, or something). I remember solving a ...
5
votes
1answer
151 views

Can the following probabilistic argument about eigenvalues be made rigorous?

Consider the following $n \times n$ matrix $$ \left( \begin{matrix} 1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 1/2 & 0 & 1/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 1/2 ...
2
votes
1answer
400 views

Limit of a Markov transition matrix

Here, $$T_n=\begin{pmatrix} 1&&&&&&\\ \frac{n-1}{n}&0&\frac{1}{n}&&&&\\ &\frac{n-2}{n}&0&\frac{2}{n}&&&\\ ...
2
votes
2answers
129 views

Linear Algebra: Prove there exists a state matrix

The problem: The attempt at solution: Am I supposed to just choose random numbers (like I did) just for the sake of proving that there is a solution, or is there a predetermined way of doing ...
5
votes
2answers
425 views

Logarithm of a Markov Matrix

Start with a Markov matrix $\mathbf{M}$, whose elements are all between $0 \le \mathbf{M}_{ij} \le 1$ and each row sums to one. There is a natural connection with this matrix and the rate matrix ...
5
votes
2answers
1k views

stochastic matrices?

Let $A$ be a symmetric stochastic matrix, such that the sum over the columns, for each row, is 1, and all elements are positive. $A$ dimensions are $n \times n$ Let's say that $B$ is a matrix which ...