##### Activity4.3.5

1. We will rewrite $$C$$ in terms of $$r$$ and $$\theta\text{.}$$ Explain why

\begin{equation*} \left[\begin{array}{rr} a \amp -b \\ b \amp a \\ \end{array}\right] = \left[\begin{array}{rr} r\cos\theta \amp -r\sin\theta \\ r\sin\theta \amp r\cos\theta \\ \end{array}\right] = \left[\begin{array}{rr} r \amp 0 \\ 0 \amp r \\ \end{array}\right] \left[\begin{array}{rr} \cos\theta \amp -\sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \text{.} \end{equation*}
2. Explain why $$C$$ has the geometric effect of rotating vectors by $$\theta$$ and stretching them by a factor of $$r\text{.}$$

3. Let's now consider the matrix $$A$$ from Example 8:

\begin{equation*} A = \left[\begin{array}{rr} -2 \amp 2 \\ -5 \amp 4 \\ \end{array}\right] \end{equation*}

whose eigenvalues are $$\lambda_1 = 1+i$$ and $$\lambda_2 = 1-i\text{.}$$ We will choose to focus on one of the eigenvalues $$\lambda_1 = a+bi= 1+i.$$

Form the matrix $$C$$ using these values of $$a$$ and $$b\text{.}$$ Then rewrite the point $$(a,b)$$ in polar coordinates by identifying the values of $$r$$ and $$\theta\text{.}$$ Explain the geometric effect of multiplying vectors of $$C\text{.}$$

4. Suppose that $$P=\left[\begin{array}{rr} 1 \amp 1 \\ 2 \amp 1 \\ \end{array}\right] \text{.}$$ Verify that $$A = PCP^{-1}\text{.}$$

5. Explain why $$A^kk = PC^kP^{-1}\text{.}$$

6. We formed the matrix $$C$$ by choosing the eigenvalue $$\lambda_1=1+i\text{.}$$ Suppose we had instead chosen $$\lambda_2 = 1-i\text{.}$$ Form the matrix $$C'$$ and use polar coordinates to describe the geometric effect of $$C\text{.}$$

7. Using the matrix $$P' = \left[\begin{array}{rr} 1 \amp -1 \\ 2 \amp -1 \\ \end{array}\right] \text{,}$$ show that $$A = P'C'P'^{-1}\text{.}$$

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