91Ó°ÊÓ

Show that if \(A X=\lambda X\) and \(A Y=\lambda Y,\) then whenever \(k, p\) are scalars, $$A(k X+p Y)=\lambda(k X+p Y)$$ Does this imply that \(k X+p Y\) is an eigenvector? Explain.

A person sets off on a random walk with three possible locations. The Markov matrix of probabilities \(A=\left[a_{i j}\right]\) is given by $$\left[\begin{array}{lll}0.5 & 0.1 & 0.6 \\\0.2 & 0.9 & 0.2 \\\0.3 & 0 & 0.2\end{array}\right]$$ It is unknown where the walker starts, but the probability of starting in each location is given by $$X_{0}=\left[\begin{array}{r}0.2 \\\0.25 \\ 0.55\end{array}\right]$$ What is the probability of the walker being in location 1 at time \(n=2 ?\)

Consider the quadratic form \(q\) given by \(q=-2 x_{1}^{2}+2 x_{1} x_{2}-2 x_{2}^{2}\) (a) Write q in the form \(\vec{x}^{T} A \vec{x}\) for an appropriate symmetric matrix \(A .\) (b) Use a change of variables to rewrite q to eliminate the \(x_{1} x_{2}\) term.