Exercise5

Here is another problem with approximate computer arithmetic that we will encounter in the next section. Consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 0.2 \amp 0.2 \amp 0.4 \\ 0.2 \amp 0.3 \amp 0.1 \\ 0.6 \amp 0.5 \amp 0.5 \\ \end{array}\right]\text{.} \end{equation*}
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 $$A$$ and the $$3\times3$$ identity matrix $$I\text{.}$$ Ask Sage to compute $$B = A-I$$ and find the reduced row echelon form of $$B\text{.}$$

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.

in-context