Report

1 Eigenvalues and Eigenvectors using the Power Method

Procedure:

• Initial Vector:

  • Start with the initial vector $(x_0=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]).$
  • Normalize the initial vector to have a length of 1.

• Power Method Iterations:

  • Compute successive vectors $(x_1,x_2,x_3)$ using $(x_{i+1}=A\cdot x_i)$ and normalize each time.
  • Continue iterations until the vectors agree to at least 4 decimal places.

• Calculate Dominant Eigenvalue:

  • Use the formula $(\lambda =\frac{(x\cdot A\cdot x)}{(x\cdot x)})$ with the vector at convergence. Results:

• Part (a) Computed Vectors:

  • $x_0=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]$
  • $x_1=\left[\begin{array}{cccc}0.7071& 0.7071& 0& 0\end{array}\right]$
  • $x_2=\left[\begin{array}{cccc}0.8018& 0.5345& 0.2673& 0\end{array}\right]$
  • $x_3=\left[\begin{array}{cccc}0.7182& 0.6156& 0.3078& 0.1026\end{array}\right]$

• Part (b) Convergence:

  • Number of iterations until convergence: 18
  • Last two vectors:
    • $\text{Previous vector}=\left[\begin{array}{cccc}0.3955& 0.5207& 0.7073& 0.2685\end{array}\right]$
    • $\text{Current vector}=\left[\begin{array}{cccc}0.3954& 0.5207& 0.7074& 0.2685\end{array}\right]$
  • Dominant eigenvalue: 3.6339

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2 Market Share Model using a Markov Chain

Procedure:

• Markov Process:

  • Define the transition matrix $(P=\left[\begin{array}{ccc}0.9& 0.10.3& 0.7\end{array}\right]).$
  • Start with the initial state vector $(x_0=\left[\begin{array}{cc}0.5& 0.5\end{array}\right]).$

• Calculate State Distribution:

  • Find the state distribution after four months by calculating $(P^4\cdot x_0).$

• Determine Steady-State Distribution:

  • Continue calculating $(P^k)$ until the matrices converge to a steady-state matrix.
  • Determine the steady-state distribution vector. Results:

• Part (a) State Distribution After Four Months:

  • Matrix ( P^4 ): $P^4=\left[\begin{array}{ccc}0.7824& 0.21760.6528& 0.3472\end{array}\right]$
  • State distribution ( x_4 ): $x_4=\left[\begin{array}{cc}0.5000& 0.5000\end{array}\right]$

• Part (b) Convergence:

  • Number of iterations until convergence: 17
  • Steady-State Matrix ( Q ): $Q=\left[\begin{array}{ccc}0.7500& 0.25000.7499& 0.2501\end{array}\right]$
  • Steady-state distribution vector: $\left[\begin{array}{cc}0.5000& 0.5000\end{array}\right]$