###### Activity4.3.2

Once again, we will consider the matrices

\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right],\qquad D = \left[\begin{array}{rr} 3 \amp 0 \\ 0 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}

The matrix $$A$$ has eigenvectors $$\vvec_1=\twovec{1}{1}$$ and $$\vvec_2=\twovec{-1}{1}$$ and eigenvalues $$\lambda_1=3$$ and $$\lambda_2=-1\text{.}$$ We will consider the basis of $$\real^2$$ consisting of eigenvectors $$\bcal= \{\vvec_1, \vvec_2\}\text{.}$$

1. If $$\xvec= 2\vvec_1 - 3\vvec_2\text{,}$$ write $$A\xvec$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

2. If $$\coords{\xvec}{\bcal}=\twovec{2}{-3}\text{,}$$ find $$\coords{A\xvec}{\bcal}\text{,}$$ the representation of $$A\xvec$$ in the coordinate system defined by $$\bcal\text{.}$$

3. If $$\coords{\xvec}{\bcal}=\twovec{c_1}{c_2}\text{,}$$ find $$\coords{A\xvec}{\bcal}\text{,}$$ the representation of $$A\xvec$$ in the coordinate system defined by $$\bcal\text{.}$$

4. Explain why $$\coords{A\xvec}{\bcal} = D\coords{\xvec}{\bcal}\text{.}$$

5. Explain why $$C_{\bcal}^{-1}A\xvec = DC_{\bcal}^{-1}\xvec$$ for all vectors $$\xvec$$ and hence

\begin{equation*} C_{\bcal}^{-1}A = DC_{\bcal}^{-1}\text{.} \end{equation*}
6. Explain why $$A = C_{\bcal}DC_{\bcal}^{-1}$$ and verify this relationship by computing $$C_{\bcal}DC_{\bcal}^{-1}$$ in the Sage cell below.

in-context