0
votes
0answers
18 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
0
votes
0answers
14 views

Matrix multiplier for ODE

I have matrix C with dimensions $3 \times 3 $ and it is skew symmetric too C is given by $C(0,0)=0,C(1,1)=0,C(2,2)=0 \tag 1$ $C(1,0)= sc_0+ px (c_1-c_0),C(0,1)=-C(1,0) \tag 2 $ $C(0,2)= ...
0
votes
0answers
12 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
3
votes
3answers
226 views

Solving inhomogenous ODE

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for ...
2
votes
0answers
43 views

How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
2
votes
1answer
39 views

Find $e^{AT}$ where $A$ is a Matrix that is given

How to find the value of $e^{At}$ where $A$ is the matrix $A =\begin{bmatrix} 4 & 3 \\ 2 & -1 \end{bmatrix}$
2
votes
0answers
39 views

Drawing phase portrait

This is the question in my textbook. I am a bit lost for 3 hours now. Could anyone please point me to the right direction? Let the $2 \times 2$ matrix $A$ have real, distinct eigenvalues $\lambda$ ...
1
vote
3answers
65 views

Deducing the exact solution of a ODE

In page 53 of Arieh Iserles's A first course in the numerical analysis of differential equations, he presents the following ODE: $(\vec{y})'=\Gamma\cdot\vec{y}$, $\vec{y}(0)=\vec{y_0}$ Using the ...
1
vote
0answers
58 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
4
votes
2answers
87 views

Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...
2
votes
1answer
49 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
1
vote
0answers
28 views

3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
1
vote
1answer
50 views

Differential equation of the form $y'=Ay+b(x)$ with $b(x)=(\sin{(\omega x)},0)$

I have a question regarding the following specific differential equation. $$y'=\left(\begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix}\right)y+\left(\begin{matrix} ...
0
votes
0answers
28 views

Find det M in the differential equation

Let's take a scalar equation: $\left(\tan{t}\right)y'''+2y''-\left(\sin{t}\right)y'+2y=0$ Defined on the interval $ t \in (-\frac{\pi}{2}, \frac{\pi}{2})$ Let $M(t,\frac{\pi}{4})$ be a solving matrix ...
0
votes
1answer
32 views

Proof of Lyapunov Stability for Constant Matrix System

I am trying to find the necessary and sufficient conditions for the point of equilibrium x=0 of $x'=Ax$ to be Lyapunov stable, where A is constant matrix. The book I'm using briefly touches on this, ...
4
votes
2answers
82 views

Show each eigenvalue of a companion matrix has geometric multiplicity $=1$.

Given the differential equation $$x^{(n)}(t)+c_{n-1}x^{(n-1)}(t) + \dotsb + c_1x'(t) + c_0=0,$$ we can form a vector $\xi = (x, x', \dotsc, x^{(n-1)})$, and then we have $$\xi'(t) = A\xi,$$ where $A$ ...
2
votes
0answers
29 views

Matrix differential equation MX' = AX+XB+C(t)

Here is matrix differential equation: $$ \mu \frac {dX}{dt}=AX+XB+C(t) $$ $$ X(0) = X_0 $$ Here $\mu$ is real diagonal matrix, $X$ is $m$ by $n$ matrix. $A$, $B$ are real square matrices of constant ...
1
vote
0answers
31 views

What are the different solution concepts for Matrix-Ordinary Differential Equation [Theory Question]

I was recently given a ODE to solve from a boss at work, with the knowledge that I haven't done them before and this will help me learn. I've spent 10 hours so far learning the basics of ODEs. The ...
0
votes
2answers
74 views

Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
1
vote
0answers
40 views

Finding Fundamental Matrix of Differential Equations with Periodic Coefficients

So I am supposed to find the fundamental matrix of $x'_1=(1+2\cos(2t))x_1+(1-2\sin(2t))x_2$ $x'_2=-(1+2\sin(2t)x_1+(1-2\cos(2t))x_2$ The book suggests using the transformation $x=Q(t)\hat{x}$ ...
0
votes
3answers
31 views

Determinant of solution matrix

Let $\phi(t)$ be a solution matrix. Show that $$\det\phi(t)=\det\phi(t)\exp\int_{t_0}^t\sum_{j=1}^na_{jj}(s)\,ds.$$ I know that $[\det\phi(t)]'=\sum_{j=1}^na_{jj}(t)\det\phi(t),$ but I am not how to ...
2
votes
1answer
48 views

Second Order Derivative of a function $f:R^2\to R^2$

The Exercise: My Work: Part 1: $$ Df=\left( \begin{array}{ccc} D_1f_1 & D_2f_1\\ D_1f_2 & D_2f_2 \\\end{array} \right) $$ $$f_1(x,y)=\sin x+\sin y$$ $$f_2(x,y)=\cos x+\cos y$$ $$ ...
1
vote
1answer
80 views

Find the determinant of a solving matrix

I have such ODE: $$\frac{dy}{dt}=\begin{pmatrix} \sin^2t & e^{-t} \\ e^t & \cos^2t \end{pmatrix} y=A(t)y(t)$$ and let $M(t,1)$ be the solving matrix (a matrix whose columns ...
1
vote
2answers
76 views

Why can't you swap rows in the matrix for a system of linear differential equations?

If you are given a Matrix A, and then asked to solve the initial value problem x'=Ax, why can one not swap rows before starting the problem. I tried it with a 3x3 matrix on wolfram alpha and got two ...
0
votes
1answer
26 views

Fundamental Matrix with Sums

Let $$\Phi(t)=\begin{bmatrix} x_{11}(t) & x_{12}(t)\\ x_{21}(t) & x_{22}(t) \end{bmatrix} $$ be a fundamental matrix for $$x'=A(t)$$ where $$A=\begin{bmatrix} a_{11}(t) & a_{12}(t)\\ ...
0
votes
1answer
47 views

Integrating a Matrix in differential-equations

Find $$\int_0^t A(s)ds$$ if $$A(t)=\begin{pmatrix}\sin(t),\cos(t)\\ -\sin(t),\cos(t)\end{pmatrix}$$ I'm a little confused with the format of the question because it asks me to integrate with respect ...
1
vote
1answer
42 views

number of solutions of a system of linear equations

Consider a system $\Sigma (y)$ of $m$ linear equations in $n$ variables $x_1,\cdots,x_n$: $\sum_{j=1}^n a_{i,j}(y)\cdot x_j=b_i(y)$, $i=1,\cdots,m$, whose coefficients $a_{i,j}(y)$ and $b_i(y)$ are ...
3
votes
2answers
73 views

how to compute the determinant of a linear map

Let $V$ be the vector space of $m\times n$ matrices over a field $F$. Fix an $m\times m$ matrix $A$ and an $n\times n$ matrix $C$, and consider the map $\phi: V\longrightarrow V$ defined by ...
0
votes
1answer
78 views

Fundamental matrix for a given system of equation

Question is to find the fundamental matrix(F(t)) satisfying F(0)=I for the given system of equation below. $$ x' =\left(\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right)x $$ My solution ...
1
vote
1answer
63 views

Second Order ODE with Matrix

Consider a (unforced) mass-spring system with mass $m = 1$, spring constant $k = 1$, and damping constant $c$, so that the displacement $x(t)$ satisfi es $x'' + cx' + x = 0$. Re-write this ODE as a ...
0
votes
0answers
26 views

Existence of solutions of a Matrix Riccati equation

I have the following Matrix Riccati Equation: $\dot{X}(t)=A \cdot X(t)+X(t) \cdot B + X(t)X(t)$ where $X(t), A$ and $B$ are real square matrices but in general not symmetric. The coefficients $A$ ...
2
votes
3answers
131 views

Chain rule for matrix exponentials

I need help in proving the following theorem: If $M(t)$ is an $n \times n$ matrix of differentiable functions, then $$ \frac{d}{dt}\left( \exp(M(t))\right) = \frac{d}{dt}M(t) \exp(M(t)) = ...
1
vote
1answer
112 views

Fundamental matrix solution and commutativity.

Please I have a question. Let $$y'(t) = M(t)y(t)~~~~~~~~~~~(*)$$ where $M(t)$ is a matrix with continuous entries on the interval $(a,b)$. Let $Y(t,t_0)$ be its fudamental solution. It is known ...
0
votes
0answers
29 views

What's the differentiation of the trace of complex matrix

Condition: all the matrices are complex. $\dagger$ denotes the conjugate transpose, * denotes the conjugate, $\triangledown_a f()$ denots the differentiation of a function $f$ with respect to $a$. I ...
2
votes
1answer
80 views

Understanding how to take derivatives with matrices

Currently we are doing 2nd order differential equations (we already did systems of homogenous two first order equations) and now that we have non-homogenous 2nd order equations we are doing method of ...
6
votes
1answer
76 views

Iterative method for matrix differential equation

Let $A$ and $X(t)$ be $n\times n$ matrices. I want to solve the matrix differential equation $$\dfrac{dX}{dt}(t)=AX(t)$$ with $X(0)=I$ (the $n\times n$ identity matrix) using the Picard iterative ...
1
vote
0answers
45 views

Given $\mathbf{A}$ stable (all negative eigenvalues), produce a bound on $\|\mathbf{B}\|$ such that…

Given a system: $\dot{\mathbf{x}}=\left(\mathbf{A}+\mathbf{B}\right)\mathbf{x}$ Can you bound $\|\mathbf{B}\|$ s.t. the origin of $\mathbf{x}$ is exponentially stable using a Lyapunov function? ...
1
vote
2answers
123 views

Derivative of mixed matrix terms with inverse matrix

I've been trying to solve two matrix derivative terms including an inverse matrix but I am unable to find a clue : 1) Derivative of $KG^{-1}J$ with respect to $G$. 2) Derivative of ...
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{ } ...
0
votes
2answers
166 views

Solving Linear Systems of Differential Equations - Phase Portraits

I'm working on trying to find the phase portrait for a differential equations such that $$ A = \begin{matrix} a_1 & a_2 \\ 0 & a_3 \end{matrix}$$ so that $$x_1'(t) = a_1x_1+a_2x_2 \text { and ...
0
votes
1answer
31 views

ODE with solution in a subspace

The task is to show that if $A$ is a Markov matrix, as to say the sum of all the entries in $A$ for each column equal $0$ and all the entries $a_{ij} ≥ 0$ for $i\neq$j then the solution to the ODE ...
0
votes
4answers
107 views

Book Searching in Stability Theory.

Can anyone recommend me a book on Stability Theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I really want to understand it, ex: ...
1
vote
1answer
71 views

differentiation of vector norm

what would be the differentiation of this equation :- $$f(A) = \sum_i \left \| Y_{i} - AB_{i} \right \|^2 + \left \| A - A_\text{constant} \right \|^2$$ wrt to $A$. where $Y$ is a column vector and ...
1
vote
1answer
45 views

Solving Linear ODE using matrices

What I don't understand here is where or how the operator for this solution is formed. Shouldn't the values of the operator be A=(1,0,0,1)? (in the form a11, a12, a13, a14 respectively). Any help ...
3
votes
1answer
3k views

Integrating a matrix

Is it possible to integrate a matrix? I've been working through a problem and come up with $$\int_{t_0}^t \begin{pmatrix}\sin(s)\cdot\cos(\beta s)\\ \cos(s)\cdot\cos(\beta s)\end{pmatrix}ds$$ I'm ...
0
votes
1answer
91 views

Commuting Exponential Matrices

Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$. Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal. $A, B$ square matrices. ...
2
votes
1answer
89 views

Solving a system of Differential Equations with RREF'ed Matrix

Is there a way to solve for example: Where $D$ is the differential operator. $$(D-4)x + D^2y = 0$$ $$(D+1)x + Dy = 0$$ Without the operator $$x' - 4x + y'' = 0$$ $$x' + 1x + y' = 0$$ Using a ...
0
votes
1answer
217 views

Calculating Log-likelihood using Raphson and Jacobian matrices?

I am reading the following paper: http://www.ntuzov.com/Nik_Site/Niks_files/Research/papers/stat_arb/Ahmed_2009.pdf and in particular page 13. I want to try and calculate lambda_t(p) = exp^(Beta^T ...
2
votes
1answer
35 views

A simple question regarding $df(E)$ for $f(A)=A^{-1}$

I am trying to follow the document on http://web.mit.edu/people/raj/Acta05rmt.pdf, and I got a simple question that why for $f(A)=A^{-1}$ we have $df(E)=-A^{-1}EA^{-1}$ on page 5? (where $E$ is a ...
0
votes
0answers
112 views

Solving system of ODE by variation of constants

I have the non-homogenous system of ODEs ($x'=Ax+b$) with IVP: \begin{equation} x' = \begin{pmatrix}0&1\\-1&0\end{pmatrix}x + \begin{pmatrix}0\\t\end{pmatrix}, x(0) = ...