Due Friday, April 29.
(a) Suppose that $A$ is an $m \times m$ matrix, and $A + A^H$ is positive-definite. It follows that any eigenvalue $\lambda$ of $A$ must have a positive real part — why? (Hint: suppose $Ax=\lambda x$ and plug the eigenvector $x$ into $x^H (A + A^H) x$, which must be _____.)
(b) Consider $A = \begin{pmatrix} -1 & -3 & -3 & -7\\ 3 & -2 & -2 & -2\\ 3 & 2 & -3 & 1\\ 7 & 2 & -1 & -4\\ \end{pmatrix}$. Suppose that we multiply an arbitrary vector $y$ by $e^A$ over and over. With the help of your answer from (a), using no numerical calculation, explain why you would expect the resulting vector to grow, decay to zero, oscillate forever, or approach a nonzero steady state (pick one)?
(a) Suppose $A=X^H X$ and $B = Y^H Y$ are two positive-definite $n \times n$ matrices, for some $X$ and $Y$ matrices. $A + B = Z^H Z$ where $Z = \_\_\_$ (give an explicit formula in terms of $X$ and $Y$, no calculation required). What must be true of the dimensions of the matrices $X$, $Y$, and $Z$?
(b) Suppose $A = X^H C X$ where $C$ is positive-definite and $X$ has full column rank. Why must $A$ be positive-definite?
You are given the matrix $$ A = \begin{pmatrix} 7 & 4 & -5 \\ 4 & -2 & 4 \\ -5 & 4 & 7 \end{pmatrix} $$
(a) Two of the eigenvalues of $A$ are 6 and –6. Find the third eigenvalue. The eigenvalues must be real because $A$ is ____________. Check your answer in Julia using eigvals(A) (don't forget to do using LinearAlgebra first).
(b) Two eigenvectors of $A$ are the column vectors (1,1,1) and (1,-2,1). Find the third eigenvector. The eigenvectors must be ____________ to one another because $A$ is ____________
(c) Suppose that $x(t)$ is the solution to $\frac{dx}{dt} = Ax - 12x$ with $x(0) = (1,0,0)$. After a long time $x(t)$ is approximately parallel to what vector? Check your answer by computing $x(t) = e^{(A-12I)t} x(0)$ in Julia for a large $t$ using exp((A-12I)*t).
(d) Give an exact expression for $x(t)$ from (c). You should be able to write your answer easily, without doing Gaussian elimination or anything fancy — only a few dot products need to be computed, because the eigenvectors are ____________. Check your answer in Julia using exp((A-12I)*t).
(e) As $t \to \infty$, the matrix $e^{(A-12I)t}$ approaches what projection matrix $P$?
Suppose that $A$ is a Hermitian positive-definite matrix, and we find a sequence of vectors $x^{(1)}, x^{(2)}, x^{(3)}, \ldots$ by repeatedly solving the recurrence equation: $$ x^{(n+1)} - x^{(n)} = -A\frac{x^{(n+1)} + x^{(n)} }{2} $$ starting from a vector $x^{(0)}$.
(a) Derive a formula $x^{(n+1)} = \mbox{(some matrix)} x^{(n)}$ in terms of "some matrix" that does not depend on $n$.
(b) Your formula from (a) involves the inverse of a matrix. Why must that matrix always be invertible?
(c) If $17$ is an eigenvalue of $A$, what is an eigenvalue of your "some matrix" from (a)?
(d) Would you expect the sequence of vectors $x^{(n)}$ to grow, decay, or oscillate as $n$ becomes large?