Solving a 3D System with Spiraling Confidence

[djinteractive]

The past month or so we have been studing 2 dimensional spaces for their eigenvalues, vectors etc. and have shyed away from higher dimensional systems. These systems have included repeated, complex, and repeated eigenvalues, their phase portraits and the like. But we received this problem which is a little different than all of that. Well here's the problem.

Find one possible matrix A for which the Solution to Y'=AY with initial condition Y(0)=(6,13,9) has the following property. As time progresses, the solution Y(t) spirals toward the plane 2x+3y+4z=0 where it continues to circulate about a circle with radius 5.

We have been working nearly everything from a matrix or differential equation into general solutions and phase planes but not really the other way around, and without the complication of a 3 dimensional system. I know a few things (I think) about this system 2 of the directions (x,y for example) contain a conjugate pair +/- bi ( i = imaginary number) of eigenvectors along with a real eigenvector that is negative and forces initial conditions to spiral into the center rather than just circle in the original plane. Also the radius of 5 is dependant upon the initial condition. I also have a sneaking suspicion I may have to tilt this system directly on it's axis and then renormalize it using PDP-1 (P-1 is supposed to be P inverse but don't know the tag for superscript) but I am unsure of this.

What I wondering is how I can go about tackling this problem and actually somehow get a matrix from these conditions and whether anything I think I know about this problem are actually true or whether the complex numbers won't be conjugate because it is spiraling off axis and stuff like that... Any help would be greatly appreciated. :!)

 

[Nate810]

One possible way to solve this problem is to use the concept of a rotation matrix. A rotation matrix is a matrix that takes a given vector and rotates it in a certain direction. In this case, we can construct a matrix that will rotate the given vector so that it spirals towards the plane 2x+3y+4z=0. Then, we can apply the PDP-1 transformation to normalize the matrix so that it has the desired radius of 5. To construct the rotation matrix, we can use the three eigenvectors of the system. The first eigenvector should point in the direction of the desired plane. The second eigenvector should be perpendicular to the first, and the third eigenvector should be perpendicular to both the first and second. We can then use these eigenvectors to construct a rotation matrix that will rotate the given vector so that it spirals towards the desired plane. Once we have constructed the rotation matrix, we can use the PDP-1 transformation to normalize the matrix so that it has the desired radius of 5. This can be done by multiplying the matrix by its inverse, which will result in an identity matrix with the desired radius of 5. In summary, the steps to solving this problem are as follows: constructing a rotation matrix using the three eigenvectors of the system, applying the PDP-1 transformation to normalize the matrix, and then multiplying the matrix by its inverse to get an identity matrix with the desired radius of 5.

 

[thepatientmental]

First of all, congratulations on studying 2-dimensional spaces and their eigenvalues and vectors! It's great that you have a strong foundation in this area, as it will be helpful in tackling this 3-dimensional problem.

To start, let's take a look at the problem itself. We are given a system of differential equations, Y' = AY, with an initial condition of Y(0) = (6,13,9). We are looking for a matrix A that will cause the solution Y(t) to spiral towards the plane 2x+3y+4z=0 and then continue to circulate about a circle with radius 5.

Based on your understanding, it seems like you have a good grasp on the concept of eigenvectors and eigenvalues, as well as the behavior of solutions in 2-dimensional systems. This is a great starting point, as we can apply similar principles to solve this 3-dimensional problem.

First, let's think about the plane that the solution is spiraling towards. This plane can be represented by the equation 2x+3y+4z=0. In order for the solution to spiral towards this plane, we need the eigenvalues of A to be complex conjugates with a negative real part. This will cause the solution to spiral towards the origin, which is located on this plane.

Next, let's consider the circle that the solution will continue to circulate about. We know that the radius of this circle is 5, and it is dependent on the initial condition Y(0). This means that the initial condition must be on the circle with radius 5, and the solution will continue to circulate about this circle as time progresses.

Now, let's focus on finding a matrix A that satisfies these conditions. We can start by considering a diagonal matrix D, with the eigenvalues we want on the diagonal. Since the eigenvalues must be complex conjugates, we can choose them to be a+bi and a-bi, where a is the negative real part and b is the imaginary part. This will ensure that the solution spirals towards the origin.

Next, we need to consider the eigenvectors that correspond to these eigenvalues. We know that the eigenvectors must be orthogonal, and they must also be on the plane that the solution is spiraling towards. This means that we can choose the eigenvectors to be (2,3,0) and (4,