Science:Math Exam Resources/Courses/MATH307/April 2006/Question 02 (a)

MATH307 April 2006

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Question 02 (a)

Consider the diagonalizable matrix $A={\begin{bmatrix}1&0\\1&0.5\end{bmatrix}}={\begin{bmatrix}1&0\\2&1\end{bmatrix}}{\begin{bmatrix}1&0\\0&0.5\end{bmatrix}}{\begin{bmatrix}1&0\\2&1\end{bmatrix}}^{-1}$ (a) Find a solution $x(t)={\begin{bmatrix}x_{1}(t)\\x_{2}(t)\end{bmatrix}}$ of the difference equation $\displaystyle x(t+1)=Ax(t)$ satisfying the initial condition $x_{0}={\begin{bmatrix}4\\1\end{bmatrix}}$. Determine the limit of the ratio $\displaystyle x_{1}(t)/x_{2}(t)$ as $t\to \infty$.

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We want to find the ratio of $\displaystyle x_{1}(t)/x_{2}(t)$ as $t\to \infty$ for the difference equation $\displaystyle x(t+1)=Ax(t)$. The difference equation suggests that for every increment of 1 unit time, we want to multiply $\displaystyle x(t)$ by $\displaystyle A$. The matrix A is already given in the diagonalized form of $A={\begin{bmatrix}1&0\\2&1\end{bmatrix}}{\begin{bmatrix}1&0\\0&0.5\end{bmatrix}}{\begin{bmatrix}1&0\\2&1\end{bmatrix}}^{-1}$ Given the initial condition of $x_{0}={\begin{bmatrix}4\\1\end{bmatrix}}$, we have the equation $\displaystyle x_{k}=A^{k}x_{0}$, which can also be expressed as the form $x_{k}=c_{1}(1)^{k}{\begin{bmatrix}1\\2\end{bmatrix}}+c_{2}(0.5)^{k}{\begin{bmatrix}0\\1\end{bmatrix}}$ where you can obtain c1, c2 by plugging in the initial condition x0 and solving $x_{0}={\begin{bmatrix}4\\1\end{bmatrix}}=c_{1}{\begin{bmatrix}1\\2\end{bmatrix}}+c_{2}{\begin{bmatrix}0\\1\end{bmatrix}}$ As $k\to \infty$, the component with $0.5^{k}$ would reach 0 and you would be left with $x_{k}=c_{1}{\begin{bmatrix}1\\2\end{bmatrix}}$ and therefore the ratio of $\displaystyle x_{1}(t)/x_{2}(t)$ would just be $\displaystyle 1/2$.