Exercise 5

For each of the following matrices, use Sage to determine its eigenvalues, their multiplicities, and a basis for each eigenspace. For which matrices is it possible to construct a basis for $\real^3$ consisting of eigenvectors?

1. $A = \left[\begin{array}{rrr} -4 & 12 & -6 \\ 4 & -5 & 4 \\ 11 & -20 & 13 \\ \end{array}\right]$

2. $A = \left[\begin{array}{rrr} 1 & -3 & 1 \\ -4 & 8 & -5 \\ -8 & 17 & -10 \\ \end{array}\right]$

3. $A = \left[\begin{array}{rrr} 3 & -8 & 4 \\ -2 & 3 & -2 \\ -6 & 12 & -7 \\ \end{array}\right]$