Exercise 10

In the next chapter, we will consider certain $n\times n$ matrices $A$ and define a function $$q(\xvec) = \xvec\cdot(A\xvec),$$ where $\xvec$ is a vector in $\real^n$.

1. Suppose that $A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix}$ and $\xvec=\twovec21$. Evaluate $q(\xvec) = \xvec\cdot(A\xvec)$.

2. For a general vector $\xvec=\twovec xy$, evaluate $q(\xvec) = \xvec\cdot(A\xvec)$ as an expression involving $x$ and $y$.

3. Suppose that $\vvec$ is an eigenvector of a matrix $A$ with associated eigenvalue $\lambda$ and that $\vvec$ has length 1. What is the value of the function $q(\vvec)$?