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