Solving Complex Eigenvectors: Find Matrix A & Compute Solution

In summary: So your initial condition should be [y(0) y'(0)] = [2 0]^T, which gives you the values for c1 and c2.
  • #1

Homework Statement



The higher order equation y"+y=0 can be written as a unknown d/dt[y y']=[y' y"]=[y' -y]

If this is du/dt=Au, what is the 2x2 matrix A? Find its eigenvectors and eigenvalues, and compute the solution THAT STARTS FROM y(0)=2, y'(0)=0.

Homework Equations



y'=Ay
y(0)=y0

The Attempt at a Solution



I found matrix A
[0 1
-1 0].
The eigenvalues are i and -i, and the eigenvectors
[1 -i]^T
[1 i]^T

I found the geneal solution to be:
y(t) = c1eit[1 i]^T+c2e-it[1 -i]^T

Which is equivalent,
y(t)=c1[cos(t) -sin(t)]^T + c2[sin(t) cos(t)]^T

I just don't know how to incorporate the initial conditions that y(0)=2 and y'(0)=0?

Any ideas?
 
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  • #2
You know the function y(t), and you know what y(0) equals (and you know y'(t) from differentiating your original y(t) equation), so then try plugging in for t and see what you get.
 
  • #3
That's what I tried doing but I'm getting a funky solution:

y(0)=2=c1*[1 0]^T + c2*[0 1]^T, so this is saying that [c1 c2]^T=2 ?

When I differentiate y(t) as in previous post,
y'(t)=[c2*cos(t)-c1*sin(t) -c1*cos(t)-c2*sin(t)]^T
y'(0)=0=[c2 -c1]^T=0

Totally lost!
 
  • #4
Hi tatianaiistb! :smile:

Your solution is not y(t), but [y(t) y'(t)] which is a vector as it should be.
 

1. What are complex eigenvectors and why do we need to solve them?

Complex eigenvectors are vectors that have complex components (i.e. with imaginary numbers). They arise when solving systems of linear equations involving complex numbers. We need to solve for complex eigenvectors because they represent important properties of a matrix, such as its eigenvalues and diagonalizability, which have applications in various fields of science and engineering.

2. How do we find matrix A given a set of complex eigenvectors?

To find matrix A, we can use the formula A = XΛX^-1, where X is the matrix containing the complex eigenvectors as its columns, and Λ is the diagonal matrix containing the corresponding eigenvalues. This formula is derived from the eigenvalue equation Av = λv, where v is the eigenvector and λ is the eigenvalue.

3. Can we use the same method to solve for complex eigenvectors as we do for real eigenvectors?

Yes, the method for solving complex eigenvectors is the same as that for real eigenvectors. We can use the characteristic equation det(A-λI) = 0 to find the eigenvalues, and then solve for the corresponding eigenvectors using the eigenvalue equation Av = λv.

4. Is there a limit to the number of complex eigenvectors a matrix can have?

Yes, a square matrix can have at most n complex eigenvectors, where n is the size of the matrix. This is because a diagonalizable matrix can have at most n distinct eigenvalues, and each eigenvalue can have at most one corresponding eigenvector.

5. Can we use complex eigenvectors to solve for the inverse of a matrix?

Yes, complex eigenvectors can be used to find the inverse of a matrix. If a matrix A has a set of complex eigenvectors, we can use the formula A^-1 = XΛ^-1X^-1, where X is the matrix containing the complex eigenvectors, and Λ^-1 is the diagonal matrix containing the reciprocals of the eigenvalues. This formula is derived from the fact that XΛX^-1 is the diagonalization of A.

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