Activity 7.1.2

Remember that the Sage command A.right_eigenmatrix() attempts to find a basis for $\real^m$ consisting of eigenvectors of $A$. In particular, the assignment D, P = A.right_eigenmatrix() provides a diagonal matrix $D$ constructed from the eigenvalues of $A$ with the columns of $P$ containing the associated eigenvectors.

1. For each of the following matrices, determine whether there is a basis for $\real^2$ consisting of eigenvectors of that matrix. When there is such a basis, form the matrices $P$ and $D$ and verify that the matrix equals $PDP^{-1}$.

   1. $\begin{bmatrix} 3 & -4 \\ 4 & 3 \end{bmatrix}$.

   2. $\begin{bmatrix} 1 & 1 \\ -1 & 3 \end{bmatrix}$.

   3. $\begin{bmatrix} 1 & 0\\ -1 & 2 \end{bmatrix}$.

   4. $\begin{bmatrix} 9 & 2 \\ 2 & 6 \end{bmatrix}$.

2. For which of these examples is it possible to form an orthogonal basis for $\real^2$ consisting of eigenvectors?

3. For any such matrix $A$, find an orthonormal basis of eigenvectors and explain why $A=QDQ^{-1}$ where $Q$ is an orthogonal matrix.

4. Finally, explain why $A=QDQ^T$ in this case.

5. When $A=QDQ^T$, what is the relationship between $A$ and $A^T$?