Exercise 5

Here is another problem with approximate computer arithmetic that we will encounter in the next section. Consider the matrix $$A = \left[\begin{array}{rrr} 0.2 & 0.2 & 0.4 \\ 0.2 & 0.3 & 0.1 \\ 0.6 & 0.5 & 0.5 \\ \end{array}\right]\text{.}$$

1. Notice that this is a positive stochastic matrix. What do we know about the eigenvalues of this matrix?

2. Use Sage to define the matrix $A$ using decimals such as 0.2 and the $3\times3$ identity matrix $I$. Ask Sage to compute $B = A-I$ and find the reduced row echelon form of $B$.

3. Why is the computation that Sage performed incorrect?

4. Explain why using a computer to find the eigenvectors of a matrix $A$ by finding a basis for $\nul(A-\lambda I)$ is problematic.