Due Friday, November 18 at 11am.
In class, we saw that $o = [1,1,\ldots,1,1]$ is an eigenvector of $M^T$ with eigenvalue $\lambda = 1$ for any Markov matrix $M$.
(a) If $x_k$ is an eigenvector of $M$ ($M x_k = \lambda_k x_k$) for any other eigenvalue $\lambda_k \ne 1$ of $M$, show that we must have $o^T x_k = 0$: it must be orthogonal to $o$. (Hint: use $o^T = o^T M$.)
(b) Check your result from (a) numerically for a random $5 \times 5$ Markov matrix M = rand(5,5); M = M ./ sum(M, dims=1), with eigenvalues eigvals(M) and eigenvectors X = eigvecs(M). (Do using LinearAlgebra to get eigvecs and eigvals.)
(Note: if you have a long vector v, Julia only shows a few elements by default, but you can show all the elements with @show v. You can also look at the absolute values of the elements with abs.(v), which can be easier to read than complex numbers in checking that entries are small.)
(c) If we expand an arbitrary $x$ in an eigenvector basis $x = c_1 x_1 + \cdots + c_m x_m$, letting $x_m$ be a steady-state eigenvector ($\lambda_m = 1$) and supposing all of the other eigenvalues are $\ne 1$, show that $o^T x$ gives us a simple formula for $c_m = \_\_\_\_\_\_\_\_$.
(d) Hence, if all other eigenvalues have magnitude $<1$, then $M^n x \to \_\_\_\_\_\_\_\_$ (simple formula in $o,x,x_m$) as $n \to \infty$. Check this formula against M^100 * [1,2,3,4,5] for your M from (b).
From Strang, section 6.2. Consider $A = \begin{pmatrix} 0.6 & 0.4 \\ 0.4 & 0.6 \end{pmatrix}$ and $B = \begin{pmatrix} 0.6 & 0.9 \\ 0.1 & 0.6 \end{pmatrix}$. For this problem you keep in mind the diagonalization of matrices like $A$ and $B$.
(a) Which of $A^n$ or $B^n$ (or both, or neither) go $\to \begin{pmatrix}0 & 0 \\ 0 & 0 \end{pmatrix}$ as $n \to \infty$? Double-check your answer by looking at $A^{100}$ and $B^{100}$ in Julia — these are approaching matrices of rank ________ and ________, respectively.
(b) For what values of the real scalar $\mu$ is $\sqrt{A - \mu I}$ a real matrix? Check your answer by trying sqrt(A - μ*I) for a few values of μ in Julia (do using LinearAlgebra to get I).