Exercise 9

In the next chapter, symmetric matrices---that is, matrices for which $A=A^T$---play an important role. It turns out that eigenvectors of a symmetric matrix that are associated to different eigenvalues are orthogonal. We will explain this fact in this exercise.

1. Viewing a vector as a matrix having one column, we may write $\xvec\cdot\yvec = \xvec^T\yvec$. If $A$ is a matrix, explain why $\xvec\cdot (A\yvec) = (A^T\xvec) \cdot \yvec$.

2. We have seen that the matrix $A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ has eigenvectors $\vvec_1=\twovec11$, with associated eigenvalue $\lambda_1=3$, and $\vvec_2 = \twovec{1}{-1}$, with associated eigenvalue $\lambda_2 = -1$. Verify that $A$ is symmetric and that $\vvec_1$ and $\vvec_2$ are orthogonal.

3. Suppose that $A$ is a general symmetric matrix and that $\vvec_1$ is an eigenvector associated to eigenvalue $\lambda_1$ and that $\vvec_2$ is an eigenvector associated to a different eigenvalue $\lambda_2$. Beginning with $\vvec_1\cdot (A\vvec_2)$, apply the identity from the first part of this exercise to explain why $\vvec_1$ and $\vvec_2$ are orthogonal.