2
votes
1answer
30 views

System (in a 6x6 matrix) of ordinary differential equations

One must find general solution for $$y' = \left(\begin{matrix} 1&2&-1&-2&1&2\\ -1&-2&1&2&-1&-2\\ 2&4&-2&-4&2&4\\ ...
0
votes
0answers
25 views

Quaternion Integration - Initial value problem

We have a standard form of quaternion integration equation $$ q(t) = q(t_0) \exp\left(\frac 12 \int_{t_0}^t \mathbf{\omega}(\tau) d\tau\right),\tag 1 $$ For reference you can check equation (42) in ...
0
votes
0answers
28 views

Quaternion Equivalence

Assume $R_{3\times3}$ is a rotation matrix. Question Is it true that there exists two quaternions representing this same rotation matrix $R_{3\times3}$ ? Hint : $\theta = \arccos\left( ...
0
votes
0answers
34 views

What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
1
vote
0answers
58 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
2
votes
1answer
14 views

When is the solution to a n initial value problem matrix differential equation invertible?

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
0
votes
0answers
25 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
0
votes
0answers
41 views

Transpose/multiplication of 3D matrices

I have $A(p)=\begin{bmatrix}p_1 &p_2 & p_3\\ 2p_1 &2p_2^2 & 4p_3^3\\ 3p_1 &3p_1 & 10\\ \end{bmatrix}\tag 1$ $ p= {\left(\begin{array}{c}p_1\\p_2\\p_3\\p_4 ...
0
votes
0answers
22 views

Matrix Algebra - Linear dependency

We have a given equation $ \frac{\mathrm{d}R(t) }{\mathrm{d} t}=R(t) \{(1-t)U_0+t U_1\}\tag 1$, all variables except scalar variable 't' has dimension $3 \times 3$. Given data $R(t)$ is ...
1
vote
1answer
52 views

Matrix - Commutative property

I have a rotation matrix represented as $R(t)=e^{B(t)},\tag 1$ where $B(t)$ is a skew symmetric matrix (since any rotation matrix can be expressed as a matrix exponent of a skew symmetric matrix), ...
0
votes
0answers
28 views

Diagonalize Complex ODE

I'm trying to solve for the dynamics of one coordinate of a coupled system of linear differential equations with complex coefficients. Physically, a number of single-pole harmonic oscillators with ...
0
votes
1answer
30 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
0
votes
1answer
61 views

Matrix-valued differential equation $A'(t)=A(t)B(t)$

How to solve matrix-valued differential equations of type $$A'(t)=A(t)B(t) \tag 1$$ All the given functions are square matrices of dimension $3$ and only $A(t)$ is invertible (not $B(t)$ or ...
1
vote
0answers
64 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
1
vote
0answers
54 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
3
votes
3answers
85 views

Conversion of rotation matrix to quaternion

We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
0
votes
0answers
30 views

Tool for differentiating a matrix and solving a matrix equation (in closed-form)?

Is there any software or online tool that can analytically solve a complex matrix equation, such as the following one in closed form (not numerically)? $$\frac{\partial \; ...
2
votes
0answers
17 views

using ode45 for descriptive forms

Using Matlab what would be the most efficient way to solve, $A_1x'(t) = A_2x(t)$, where both $A_1$ and $A_2$ are $n\times n$ matrices. Both are sparse matrices and hence I want to avoid inversion. ...
2
votes
0answers
39 views

Finding a solution basis of differential equation

Find a solution basis of $$y'=\left[ \begin{matrix}3&-4&-2\\2&-3&-2\\0&0&1\\ \end{matrix} \right]y \,\text{ and find the solution } \Phi \text{ with } \Phi(0) = (1,1,1).$$ I'm ...
2
votes
0answers
80 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
0
votes
1answer
49 views

First-order linear differential equation for matrix valued functions of size $3\times 3$

I have two matries given by (M' means derivative w.r.t x) $ M=\left( \begin{array}{ccc} f_1(x) & f_2(x) & f_3(x) \\ f_4(x) & f_5(x)& f_6(x) \\ f_7(x) & f_8(x) & ...
0
votes
0answers
27 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
17 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
14 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
302 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
51 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
48 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
41 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
67 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
80 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
95 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
54 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
54 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
30 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
36 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
91 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
31 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 ...
2
votes
0answers
36 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
84 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
44 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
32 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
49 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
81 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
79 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
49 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
43 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
91 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 ...