(Solved): Need help answering these questions in linear algebra related to Triangularisation with an orthogona ...

Need help answering these questions in linear algebra related to Triangularisation with an orthogonal matrix

Triangularisation with an orthogonal matrix Example \( 7.9 \) in the Study Guide (pages 21-23 of Topic 7) shows the triangularisation procedure for a matrix. Consider the following matrix \( A \), which also has eigenvalues 1,1 and 5 . \[ A=\left[\begin{array}{ccc} 3 & 2 & -2 \\ 1 & 2 & -1 \\ -1 & -1 & 2 \end{array}\right] \] i. Construct a matrix \( S \) such that it is an orthogonal matrix with the first column corresponding with the eigenvector \( \left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right] \). Calculate \( S^{-1} A S \). ii. Show that for the resulting matrix, \( S^{-1} A S=\left[\begin{array}{ll}1 & \mathbf{b}^{\top} \\ \mathbf{0} & A_{1}\end{array}\right] \), the \( 2 \times 2 \) matrix \( A_{1} \) has eigenvalues 1 and 5 by determining and then simplifying its characteristic equation. iii. Find the eigenvector from \( A_{1} \) corresponding with \( \lambda=1 \) and then construct an orthogonal \( 2 \times 2 \) matrix \( Q \) where the first column is based on your eigenvector. Hence construct the matrix \( R=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & Q \\ 0 & \end{array}\right] \) iv. Calculate \( P=S R \). Show that \( P \) is an orthogonal matrix and verify that \( P^{-1} A P \) is upper triangular.