User:Egm6321.f12.team4.hong/Report5

Problem R5.4 – An Example of a Matrix Exponential
sec20-5

Statement
Show the process of equation (1) and (2)

Solution
We know that matrix A with real value components can always be in the following form,

Comparing this with equation (1), we can say that,

 (a) Finding Eigenvalues 

Let's say that we only know matrix A and we need to find eigenvalues and eigenvectors. Then we have,

From the above matrix we can have the following relations,

In order to hold equation 5-4-6 to be true, either $$ \Phi $$ has to be a zero matrix or $$A - \Lambda$$ has to be linearly independent. Since we want $$ \Phi $$ to be non-trivial we can come up with the following equation.

Now we can solve for eigenvalues using the above equation.

(b)Finding Eigenvectors Let $$ \phi_1 = \begin{bmatrix} x_1 \\ y_1\end{bmatrix} $$ and $$ \phi_2 = \begin{bmatrix} x_2 \\ y_2\end{bmatrix} $$

If we let $$ x_1 = 1 $$ then $$ y_1 = -i $$

In the same way, we can find $$\phi_2$$

If we let $$ x_2 = 1 $$ then $$ y_2 = i $$

Finally we have the following eigenvalue matrix and its inverse.

Finally, we came up with equation (1) which is,

(c) Taking the exponential of A

From R5.2 and R5.3 we already know that,

Therefore we can write the equation as below and carry out the matrix multiplication.

Recall: Euler's Formula

If we apply Euler's Formula we now have the following.

Author and References

 * Solved and Typed by -- Seong Hyeon Hong and Kaitlin Harris
 * Reviewed by --

Problem R5.10 – Plotting Hypergeometric Function
sec64-9b

Statement
Use matlab to plot F(5, 10; 1; x) near x = 0 to display the local maximum (or maxima) in this region.

Show that

Solution
(a) Matlab Code 

(b) Matlab Result 

(c) Graph 

Author and References

 * Solved and Typed by -- Seong Hyeon Hong and Kaitlin Harris
 * Reviewed by --