0
votes
1answer
29 views

Neighbourhood of a matrix

Sometimes I find definitions which say that something happens in a neighbourhood of a matrix. For example a dynamical system generated by $x'=Ax, \ A \in \mathcal{M}(n)$ is structurally stable if ...
7
votes
1answer
276 views

Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$ a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right). $$ The first ten terms are: $0.75$ ...
1
vote
0answers
50 views

iterates of generalized matrix system

This may be somewhat of an underspecified question but I'll nonetheless give it a try. In the context of applied work, I've recently come across systems of the form $$ ...
0
votes
1answer
79 views

Jordan canonical forms and diagonalizing.

In my dynamical systems, we are asked to find the Jordan Canonical form of the Jacobian in order to analysis the linear stability at fixed points in a second order system. I believe that even for one ...
0
votes
1answer
34 views

Can I always put a system in modal form?

Given the transfer function of a system, can I always put the system in modal form? Are there exceptions?
-1
votes
1answer
48 views

is the system exponentially stable? uniformly stable?

Consider the state equation: $$ \frac{\partial}{\partial t}x(t)= A(t)x(t), \: x(\tau)=x_0 $$ $$ A(t) = \begin{pmatrix} -1 & k(t) \\ 0 & -1 & \\ \end{pmatrix}, ...
0
votes
1answer
40 views

Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?

I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!! Show that if $A(t)$ is partitioned as $$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ ...
0
votes
1answer
197 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
2
votes
1answer
119 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
1
vote
1answer
40 views

Properties for internal stability of a discrete-time system

These are two parts of a larger proof I'm working on, can't figure how i) implies ii) though. Dynamic system: $x_{(k+1)} = Ax_{k}, x(0)=x_0$ Where $A \in \mathbb{R}^{n\times n} $ is a real ...
2
votes
1answer
104 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
1
vote
1answer
73 views

Are there any mathematical/physical concepts or theories for dealing with a matrix in which the values are changing in a certain way?

As a matter of fact, my application scenario is a recommender system in which the interests/preferences of the users change. I have such a global user-interest matrix: the rows are the records of many ...
2
votes
1answer
113 views

If $P$ is an invertible transition probability matrix, does $P^{-1}[i,j]$ have any interesting meaning?

Suppose we have a Markov chain transition probability matrix $P$ that is invertible, i.e., $P^{-1}$ exists. Question: Does there exist a meaningful interpretation of the $(i,j)$ entry in $P^{-1}$? ...
4
votes
1answer
111 views

Trying to prove a matrix is always convergent.

I have a matrix $Z$ of the form $Z = \left[Q^{-1}-Q^{-1}A^T\left(AQ^{-1}A^T\right)^{-1}AQ^{-1}\right]\Phi$ where, $\Phi$ is a diagonal matrix of real non-negative values. $\Theta$ (not ...
0
votes
1answer
54 views

Matrix expansion does not decrease norms

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean ...
0
votes
1answer
76 views

anomalous minus sign in commutator of vector fields

Define $$X_A(x):=A^i_{\ j}x^j$$ where $A$ is a matrix. Why is there a minus sign in the following formula? $$[X_A,X_B]=-X_{[A,B]}$$ Edit: perhaps the question is not well posed, since what I really ...
3
votes
1answer
93 views

Repeated Eigenvalues 2

Two problema from Differential Equations; Dynamical Systems, and an Introduction to Chaos (Morris W. Hirsch,Stephen Smale.Robert L. Devaney). Examples (pages 112-113): If $$A= \begin{pmatrix} ...
2
votes
1answer
95 views

Repeated Eigenvalues

Two problems from Differential Equations; Dynamical Systems, and an Introduction to Chaos (Morris W. Hirsch,Stephen Smale.Robert L. Devaney), examples page 112,113: If $$A= \begin{pmatrix} 2 ...
2
votes
0answers
52 views

How to compress a linear operator and have the lossless composition property.

Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
4
votes
0answers
45 views

Definition and some elementary properties of the “vector turn map”

This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
3
votes
3answers
130 views

Is this map a known one?

Let $A$ be a $2\times2$ real matrix, then define $f:S^1\to S^1$ by $f(\phi)=\arctan(A\cdot(\cos \phi,\sin \phi))$. This can be viewed as a discrete dynamical system on $\mathbb{S}^1$ and I am trying ...
0
votes
1answer
48 views

$Az=λz$ lead to $x(t) = c_1*e^{\lambda_1 t}z_1+c_2*e^{\lambda2 t} z_2+…+c_p*e^{\lambda_p t}z_p$ is a solution to $dx/dt=Ax$. Why?

I'm studying a course in dynamical systems. It's a pretty much linear algebra intensive course, and it's been a while since I did that sort of things. In it, they say that if vector $z$ satisfies ...
1
vote
3answers
126 views

conditions under which real-matrix exponential are equivalent

Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$? Thanks!
0
votes
1answer
79 views

Solution to system of difference equations with repeated unit roots

Can anyone provide the forms of the solutions for the homogeneous part and particular solutions for a non-homogeneous system of two linear autonomous difference equations ...
0
votes
2answers
283 views

Closed form solution of this second order linear difference equation?

$$ y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k $$ Transform into a system of $n$ first order equations (Step 1) $$\begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align}$$ It follows that ...
0
votes
2answers
103 views

Canonical form of a Matrix question involving a conjugacy

How do i find the canonical form of this matrix, my attempt is to use it in a conjugacy for flow. $$A=\begin{pmatrix} 0&1&0 \\ -1&0&0\\ 1&1&1\end{pmatrix}$$ do i need to ...
2
votes
2answers
87 views

How do I know if a discrete time-invariant homogeneous dynamic system will reach, at some point, an equilibrium point?

Is this even possible? Given a time-invariant homogeneous dynamic system: $$x(k+1) = Ax(k)$$ My textbook defines an equilibrium point of the system as: A vector $\bar x$ is an equilibrium point ...
1
vote
1answer
82 views

Flow of D.E what is the idea behind conjugacy?

I got some kinda flow issue, ya know? well enough with the bad jokes let A be a 2x2 matrix, T a change of Coordinate matrix, and $B=T^{-1}AT$ the canonical matrix ascoiated with A. Show that the ...
1
vote
2answers
401 views

Changing parameters in a 3x3 Matrix trying to find the general solution.

Consider the system $$X^{'}= \begin{pmatrix} 0&0&a \\ 0&b&0\\ a&0&0 \end{pmatrix}X$$ depending on the two parameters a and b. 1) find the general solution of this system. ...
1
vote
1answer
114 views

e'ing a matrix and then finding eigenvalues and a ivp to converge to the origion

The question is as follow's consider the Matrix A $$A = \begin{pmatrix} \lambda & 0 &0\\ 1 & \lambda&0 \\ 0&1& \lambda \end{pmatrix},$$ Compute $e^{tA}$, and use it to ...
1
vote
1answer
35 views

Change of basis (to a more structured one) in a dynamic system, wrong result?

Given a time-invariant homogeneous dynamic system $x(k +1) = Ax(k)$, where the system matrix is: $$ A = \begin{bmatrix} 1 & -1 \\ 2 & 4 \\ \end{bmatrix} $$ ...
2
votes
0answers
76 views

Kalman filter implementation question

I have the following code to define a Kalman filter: ...
2
votes
3answers
256 views

Quadratic equation with matricial coefficients

If I have a equation in the form $${\lambda ^2}{I_N} + \lambda {M_1} + {M_2} = {0_N}$$ where ${I_N}$ is the identity matrix of order $N$, $M_1$ and $M_2$ are matrices of ($N\times N$) order and ...
1
vote
2answers
117 views

The system of $x(t+1) = Ax$ growing and retaining stability possible?

This is about general equilibrium: Suppose that $x(t)$ represents outputs of all sectors and parts of the whole economy - represented as matrix. How outputs evolve to $x(t+1)$ ...
2
votes
1answer
99 views

logarithm of a matrix base a matrix — $\mathbf{A}^x = \mathbf{B}$

I want to solve $\mathbf{A}^x = \mathbf{B}$ where $\mathbf{A}$ and $\mathbf{B}$ are both $n$-by-$n$ matrices and $x$ is real. I see that in general there may be no solutions, or multiple solutions. I ...
1
vote
1answer
105 views

Matrix-valued ODE - nonsingularity of solution

I have a matrix-valued inhomogenous linear ODE $X' = F(t)X + G(t)$, $X(0) = I_{n \times n}$, $F(t),G(t) \in \mathbb{R}^{n \times n}$, and the entries of $f$ and $g$ are continuous functions. What ...
1
vote
2answers
278 views

State transform from one state space representation to another

I have a state space representation, system S1, in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ This system is transformed with the state transform $$x=T z$$ into the system S2: $$ ...
3
votes
1answer
573 views

How to obtain a possible state space representation of this 2nd order transfer function?

I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ ...
1
vote
1answer
282 views

How to obtain the state matrix of this trajectory?

Continuous-time LTI case. I have a problem getting the state matrix of this trajectory. One element of the state matrix is known. $$ A = \begin{pmatrix} a & 4 \\c & d \end{pmatrix} $$ I ...