Exercise 4

Consider the matrices $$A = \left[\begin{array}{cc} 3 & 2 \\ -5 & -3 \\ \end{array}\right], \qquad B = \left[\begin{array}{cc} 5 & 7 \\ -3 & -4 \\ \end{array}\right]\text{.}$$

1. Find the eigenvalues of $A$. To which of the six types does the system $\xvec_{k+1}=A\xvec_{k}$ belong?

2. Using the eigenvalues of $A$, we can write $A=PEP^{-1}$ for some matrices $E$ and $P$. What is the matrix $E$ and what geometric effect does multiplication by $E$ have on vectors in the plane?

3. If we remember that $A^k = PE^kP^{-1}$, determine the smallest positive value of $k$ for which $A^k=I$.

4. Find the eigenvalues of $B$.

5. Find a matrix $E$ such that $B = PEP^{-1}$ for some matrix $P$. What geometric effect does multiplication by $E$ have on vectors in the plane?

6. Determine the smallest positive value of $k$ for which $B^k=I$.