0
votes
1answer
27 views

The largest real eigenvalue of a matrix is bigger than 1

I have a problem which is interesting: given a real matrix $A_{n\times n}$, when this matrix has a largest real eigenvalue which is strictly bigger than 1. If possible, can you give some conditions ...
2
votes
0answers
30 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
1
vote
1answer
39 views

Fixed Matrices over finite field by a map

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ ...
1
vote
1answer
42 views

Index of a function and a gradient flow

We know index of function $F:\mathbb{R}^n\to\mathbb{R}$ at critical point $x_0\in\mathbb{R}^n$ is the number of negative eigen values of Hessian matrix $DF^2(x_0)$. ...
1
vote
1answer
41 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
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 ...
0
votes
1answer
62 views

Dynamics of matrices over finite field and Similarity of matrices

Consider a set $M$ of all possible square matrices over a finite field $F_p$. Now consider a map $f_A(x)=A.x$ where $x$ $\in$ $M$ and also the matrix $A$ is a member of $M$. It is needless to mention ...
0
votes
1answer
43 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
3
votes
1answer
58 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
1
vote
1answer
34 views

What are the Routh Hurwtiz Criteria for 3$\times$3 Matrices?

The Criteria I know (for dynamical systems) is... The eigenvalues of a matrix are guaranteed to be negative if Tr($J$)<0 and det($J$)>0, where $J$ is the Jacobian of some 2 dimensional dynamical ...
0
votes
1answer
89 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 ...
1
vote
2answers
74 views

How do I Linearize

How would I solve the following problem? Linearize around the fixed points $$\left\{\begin{align}\frac{\text{d}x}{\text{d}t}&=y-x^2\\\frac{\text{d}y}{\text{d}t}&=y-x\end{align}\right.$$ I ...
0
votes
1answer
51 views

Prove that $(A,B)$ is uncontrollable $\Longleftrightarrow$ $\exists P$ $\in$ $\mathbb{R}^{nxn}$, $P \neq 0$: $PA - AP = 0$, $PB=0$

In my course advanced system Theory I had the following question: Prove the following equivalence for the pair $(A,B)$ $\in$ $\mathbb{R}^{nxn}$ x $\mathbb{R}^{nxm}$: $(A,B)$ is uncontrollable ...
1
vote
0answers
35 views

Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
0
votes
0answers
23 views

Linear Differential Equation with Quasiperiodic Coefficient: A Question

I am reading the book Synergetics. The following passage is from the book: Assume that in the differential equation $\dot{q}= a(t)q$, where $a(t)$ is quasiperiodic, i. e., we assume that $a(t)$ ...
0
votes
0answers
34 views

Find a nonhyperbolic matrix which satisfies certain conditions

A nonhyperbolic matrix A for which the planar system $\overrightarrow x' = A \overrightarrow x$ has an equilibrium point at (0,0) with the x-axis as a stable curve and the y-axis as an unstable curve. ...
2
votes
0answers
35 views

Dynamical System of Matrices

Let $T_{nm}$ be the set of all possible binary rectangular matrices of dimension $n\times m$. The cardinality of $T_{nm}$ would be $2^{nm}$. Let f be a map from $T_{nm}$ to itself. Consider a ...
1
vote
0answers
42 views

Cross Product - Moments :: Dynamics 2

This problem is related to Cross Product - Moments :: Dynamics Please look at that link for the background on the problem I am faced with right now, I have linked a pdf of the book that I am using ...
0
votes
0answers
50 views

what is the meaning/characteristics of the component-wise product of right and left eigenvectors.

I have a generic, but seemingly simple question : what is the meaning/characteristics of the component-wise product of right and left eigenvectors (for the same eigenvalue of course) ? let's call ...
0
votes
0answers
43 views

Waiting Time For Computer Cluster

There are $n$ computers. Computer users stay on their computers for a certain amount of time, $t$, throughout the day. Computer users come and go. How long will I have to wait, min/max, for a computer ...
1
vote
1answer
78 views

Linear Algebra Dynamical System Help

I was just wondering, for a dynamic system does the origin always have to be an attractor, saddle point, or repellor? Also if a matrix isn't diagonalizable then does that mean the origin cannot be a ...
4
votes
2answers
110 views

Logarithm of matrix with positive entries

For matrices with positive entries (or more generally, irreducible matrices with non-negative entries), we have the Perron-Frobenius theorem, which tells us that there will be a unique eigenvector ...
4
votes
0answers
137 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
1
vote
0answers
43 views

State Space Difference Linear Dynamic System

I am interested in finding the DIFFERENCE in the state space distributions for two linear dynamical systems (System A and System B). I am able to solve for this using the matrix exponential. But the ...
0
votes
2answers
84 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
0
votes
1answer
199 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
122 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 ...
0
votes
0answers
46 views

How is open set defined in linear map space

I got this statement in my homework: Prove that the invertible linear contractions are an open set in $Mat(2\times 2;\mathbb{R})$ I know what "invertible linear contraction" , "open set" and ...
4
votes
1answer
112 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 ...
3
votes
1answer
69 views

Uniform Hyperbolicity Decay Estimate

This question has been posted on math overflow with little interest, so I am posting it here. http://mathoverflow.net/questions/139010/uniform-hyperbolicity-decay-estimate I have been trying to ...
3
votes
1answer
187 views

Constructing a non-linear system with prerequisites about the nature of its critical points.

An exercise from the book I am reading is: "Construct a non-linear system that has four critical points:two saddle points, one stable focus, and one unstable focus." I have tried many systems. I ...
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 ...
3
votes
1answer
95 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
0answers
53 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
2answers
1k views

Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
1
vote
3answers
127 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!
1
vote
1answer
143 views

Show that $\ker W$ is $A$-invariant?

Consider the system.: $$\dot x = Ax$$ $$y= Cx $$ The observability matrix then will be $$ W= \left(\begin{matrix} C \\ CA \\ CA^2\\ \vdots \\ CA^{n-1} ...
2
votes
3answers
189 views

How to solve simple systems of differential equations

Say we are given a system of differential equations $$ \left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix} $$ Where $A$ is a $2\times 2$ matrix. ...
1
vote
1answer
205 views

Kalman filter and data extrapolation

Context of the situation: I have a system set up that can give me the position of a person in a room. I also have a light that shines on this position. However, the light are lagging behind by 0.300 ...
0
votes
1answer
85 views

How to model a system for tracking a person using kalman filter?

I need to model a system for human motion. The following link shows for to build a system for a plane. I am currently reading the documentation for a kalman filter library ...
0
votes
0answers
56 views

Studying a nonlinear system under constraints in linear fashion

Suppose that $a = aBc$ where $a$ and $c$ are vectors and $B$ is some matrix that changes as time "continuously" goes on - making this system dynamical system. But suppose that at any time, if $B$ is ...
3
votes
3answers
1k views

Poles and Zeros of Linear Systems

This period I follow a course in System and Control Theory. This is all about linear systems $$\frac{dx}{dt}= Ax + Bu $$ $$y = Cx + Du $$ where A,B,C,D are matrices, and x, u and y are vectors. ...
4
votes
1answer
527 views

Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?

Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
0
votes
1answer
96 views

Meaning of index in matrices

Question is, what does "index" mean? For systems of order greater than the number of characteristic roots of $C$ of index one Also, can anyone explain why is $u_1 + u_2 + n -1 =0$ and what ...
1
vote
2answers
118 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
145 views

Stability of a system that has (Jacobian-like) matrix with eigenvalue of less than 1 that has $x$ as non-eigenvector

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 $x(t+1)$ is ...
7
votes
2answers
206 views

Prove that Anosov Automorphisms are chaotic

Let me add some detail first. An Anosov automorphism on $R^2$ is a mapping from the unit square $S$ onto $S$ of the form $\begin{bmatrix}x \\y\end{bmatrix}\rightarrow\begin{bmatrix}a && b \\c ...
2
votes
1answer
121 views

Dynamical changing of an eigenvector

Consider a matrix $A\in\mathbb{R}^{n\times n}$. One of the eigenvalues of $A$ is zero and all the others are positive. Suppose $w\in\mathbb{R}^n$ is an eigenvector with the zero eigenvalue, i.e, ...
0
votes
1answer
278 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 ...