User:Egm6321.f12.team6.bang/report4/problem 5.3

=Report 5=

Problem Statement

 * Show $$ \displaystyle \exp [\mathbf A] = \mathbf \Phi \, \text{Diag} [\, e^{\lambda} \,, \ldots , \, e^{\lambda_n} \,] \, \mathbf \Phi^{-1} $$

Nomenclature

 * $$ \displaystyle \exp [\mathbf B] = e^{\mathbf B} $$
 * $$ \displaystyle \text{Diag} [\lambda_1,\dots,\lambda_n] = \begin{bmatrix} \lambda_1 & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \lambda_n \end{bmatrix} $$

Solution

 * Using Equation 3.1, we can transform the expression into the following.


 * Assuming that matrix A can be diagonalizable, then we can implement Equation 3.2.


 * With a property given from Equation 3.3, the equation becomes the following.


 * Using Equation 3.4, and Equation 3.5, we can derive the following expression.


 * Then, substitute Equation 3.11 expression to Equation 3.10
 * {| style="width:100%" border="0"|-

$$ \displaystyle \exp \mathbf {[A]} = \mathbf \Phi \, \text{Diag} [\, e^{\lambda_1} \,, \ldots , \, e^{\lambda_n} \,] \, \mathbf \Phi^{-1} $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |


 * (3.12)
 * }