Tagged Questions
1
vote
0answers
25 views
Diagonalizing/eigenvalues of a particular infinite dimensional matrix
I have trying to show that the continuum limit of n quantum harmonic oscillators gives rise the the klein-gordon field. However, instead of a usual finite string, I want to do it on a ring.
Assume $n ...
1
vote
2answers
27 views
Recursive formulae involving a linear operator
Given a basis $e_{1}$, $e_{2}$ in the plane, define the linear operator $F$ as $F(e_{1})=3e_{1}+e_{2}$ and $F(e_{2})=e_{2}$. Furthermore, define the sequence $u_{1},u_{2},\dots$ of vectors in the ...
1
vote
3answers
54 views
How to solve linear recurrences consisting of both $x_n$ and $y_n$?
I know how to solve a general linear, homogenous recurrence but these are the ones I've been given:(i) $x_{n+1} = 3x_n + 6y_n$(ii) $y_{n+1} = 6x_n - 2y_n$
Initial conditions: $x_0 = -1, y_0 = 0$
How ...
0
votes
0answers
37 views
qualitative results using discrete Putzer algorithm
I am studing for an exam in difference equations...the following is an exarcise from S.N.Elaydi's book.
...................................................
Define $ \displaystyle{ u_1 (n) = \lambda ...
1
vote
2answers
224 views
Solving system of recurrence relations
Base Case:
$$
\left\{
\begin{array}{c}
T(1) = 1 \\ T(2) = 1 \\T(3) = 4\end{array}
\right.
$$
I have the system:
$$
\left\{
\begin{array}{c}
T(N) = G(N-1) + F(N-1) \\
G(N) = F(N-1) + G(N-1) \\
...
0
votes
0answers
75 views
Given a state transformation matrix and a state vector, how to find the total number of changes for each individual element.
I have a vector of binary variables, $s$ representing a state at some point in time, and a transformation matrix $T$. The initial state is $s_0$.
$s_n = Ts_{n-1}$
Given a number of transformations, ...
12
votes
6answers
604 views
How to solve this recurrence relation? $f_n = 3f_{n-1} + 12(-1)^n$
How to solve this particular recurrence relation ?
$$f_n = 3f_{n-1} + 12(-1)^n,\quad f_1 = 0$$
such that $f_2 = 12, f_3 = 24$ and so on.
I tried out a lot but due to $(-1)^n$ I am not able to ...
3
votes
2answers
182 views
Understanding why the roots of homogeneous difference equation must be eigenvalues
There is some obvious relationship between the root solutions to a homogeneous difference equation (as a recurrence relation) and eigenvalues which I'm trying to see. I have read over the wiki article ...
2
votes
3answers
87 views
recurrence solution to gambler's ruin
From DeGroot 2.4.2, let $a_i$ be the conditional probability that the gambler wins all $k$ given gambler is at $i$.
$a_i = pa_{i+1} + (1 - p)a_{i-1} $
It's not clear from the text what steps are ...
0
votes
1answer
184 views
Solving Linear Systems with Singular Matrices
Good morning! For (say, homogenous) linear systems of the form
$$x_{n+1} = A x_n,$$
where $A$ is a nonsingular matrix, each initial value problem can be solved by the method of finding a general ...
1
vote
2answers
105 views
Analyzing a recurrence relation given by a Toeplitz matrix
Let $p$ be an odd prime, and $M_p$ be the $p\times p$ Toeplitz matrix over $\mathbb{F}_2$ given by $a_0=a_1=1$ and $a_{-p+1}=1$, e.g. for $p=5$ we have
$$M_5=\left[\begin{array}{ccccc}
1 & & ...
1
vote
1answer
114 views
Linear Recurrence of Matrix into Closed Form Example Explanation?
I was given the following as an example of a linear recurrence and I don't understand how it works...
Let us call the following eq1:
$$x_i = \begin{bmatrix} \sum_{z = 1}^i{zk^{z-1}} \\ (i+1)k^i \\ ...
3
votes
2answers
854 views
Linear Algebra: Finding a steady state matrix
Here is the problem:
And here is what I tried to do:
I tried doing it in my calculator and got that the answer is 40 and 160 as you keep multiplying PX. Can anyone point out what I'm doing wrong ...
3
votes
3answers
207 views
Common theory for Linear equations, linear ODEs and linear recurrence relations
In Linear equations, linear ODEs and linear recurrence relations, when solving homogeneous equations, there are a subspace of solutions, and when solving inhomogeneous equations, a particular ...
3
votes
3answers
286 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?
