This short pset is due Wednesday, November 9, since Friday is a holiday.
Suppose that $A$ is a $3\times 3$ matrix with eigenvalues $\lambda_1 = 1, \lambda_2 = -1, \lambda_3 = 2$ and corresponding eigenvectors $x_1, x_2, x_3$.
(a) Give eigenvectors and eigenvalues of $(A^2 - 3A + 4I)^{-1}$
(b) For what value(s), if any, of the scalar $\mu$ is $B = A^2 - 3A + \mu I$ singular, which corresponds to $B$ having one or more eigenvalues equal to _______.
(c) $A^n x$ for large $n$ is very nearly parallel to _______ unless $x$ is _______.
Run using LinearAlgebra followed by running eigvals(randn(3,3)) a few times.
Explain why a $3\times 3$ real matrix $A$ must have at least one real eigenvalue, and in fact this must be true for any $m \times m$ real matrix if $m$ is ______ (most general answer).
Consider the following recurrence: $$ f_n = \frac{f_{n-2} - f_{n-1}}{2} $$
Suppose that we start it with $f_0=0$ and $f_1=1$. Then the first few terms in the sequence are: $$ f_0,f_1,f_2,\ldots = 0, 1, -\frac{1}{2}, \frac{3}{4}, -\frac{5}{8}, \frac{11}{16}, -\frac{21}{32}, \frac{43}{64}, -\frac{85}{128}, \frac{171}{256}, -\frac{341}{512}, \frac{683}{1024}, -\frac{1365}{2048}, \frac{2731}{4096}, \ldots $$
(a) Similar to the Fibonacci sequence from class, we can write this recurrence in matrix form: $$\begin{pmatrix} f_n \\ f_{n-1} \end{pmatrix} = A \begin{pmatrix} f_{n-1} \\ f_{n-2} \end{pmatrix}$$ for what matrix $A$?
(b) Therefore, write a formula for $f_n = \_\_^T A^n \_\_$: fill in the blanks with two column vectors.
(c) Find the eigenvalues of $A$ and corresponding eigenvectors.
(d) From (c), what should the ratio $f_n / f_{n-1}$ approach for large $n$? Check that your prediction matches what the terms in the sequence above appear to be doing.
(e) Give an exact formula for $f_n$, using (b) and writing ____ in the basis of ____.