Laplace Transforms Homework: Initial Displacements & Velocities

[jake96]

So now that I have my eigenvalues and eigenvectors. I believe I now use them for the exponential matrix and that should give me my sin and cos equations of x₁ and x₂ from earlier in the question. However I am unsure how I am able to reach this point as I have not fully learned this yet and am unable to find suitable sources. Does anyone know of a decent source?

 

[Chestermiller]

Are you familiar with $e^{iθ}=cosθ+isinθ$? If the eigenvalues and eigenvectors were real, would you know how to proceed next?

Chet

 

[jake96]

ive only ever used it for 2x2 matrix's, I assume this way I would receive 4 equations with 4 unknowns? I'm still a little unsure as to how those unknowns are calculated, will the 4 equations be equal to the 4 y equations that Ray Vickson listed?

 

[jake96]

also I just want to double check, if I had the eigenvalue of 'i' and the corresponding eigenvector (-i,1,-i,1)
the first term of the first y equation would be -i(acost+ibsint)
with a and b as unknowns
and so on with each value and vector

 

[jake96]

Are you familiar with $e^{iθ}=cosθ+isinθ$? If the eigenvalues and eigenvectors were real, would you know how to proceed next?

Chet

OK, so I believe I have gotten my four y equation out with 4 unknowns c₁, c₂, c₃ and c₄. What will these 4 equations equal so that I am able to determine the value of each 'c'

 

[Ray Vickson]

So now that I have my eigenvalues and eigenvectors. I believe I now use them for the exponential matrix and that should give me my sin and cos equations of x₁ and x₂ from earlier in the question. However I am unsure how I am able to reach this point as I have not fully learned this yet and am unable to find suitable sources. Does anyone know of a decent source?

Check out the concept of "matrix function". Basically, if $f(x) = c_0 + c_1 x + c_2 x^2 + \cdots$ is an analytic function of $x$, we can define $f(A) = c_0 I + c_1 A + c_2 A^2 + \cdots$ for an $n \times n$ matrix $A$. Here, $I = n \times n$ unit matrix. It is a fact that for a matrix with $n$ distinct eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ we can find $n$ matrices $E_1, E_2, \ldots, E_n$ such that
$$f(A) = E_1 f(\lambda_1) + E_2 f(\lambda_2) + \cdots + E_n f(\lambda_n)$$
The crucial point is that the matrices $E_1, E_2, \ldots, E_n$ are independent of the function $f$. That means we can find them by applying this formula successively to the functions $f_0(x) = 1 = x^0, f_1(x) = x, f_2(x) = x^2, \ldots, f_{n-1}(x) = x^{n-1}$ using the known (or computable) matrices $I = A^0, A, A^2, \ldots, A^{n-1}$.

If you have repeated eigenvalues the formula becomes a bit more complicated.

See, eg., http://www.siam.org/books/ot104/OT104HighamChapter1.pdf or
Gantmacher, "Theory of Matrices, Vol. I", Chapter 5; now available free as http://www.maths.ed.ac.uk/~aar/papers/gantmacher1.pdf .

 

[jake96]

thanks for your help everyone, I've managed to get the correct answer using a 2x2 matrix made from the original equations