Due Wednesday, December 7 at 11am. This is the last pset in 18.06 this term and the last pset covered on exam 3.
For the following parts, use one (or more) of the equivalent properties (from class) that define positive-definiteness. (There are multiple possible approaches.)
(a) If $A$ and $B$ are positive-definite $m \times m$ matrices, why must their sum $A + B$ be positive definite?
(b) If $B$ is a positive-definite $m \times m$ matrix and $C$ is an $m \times n$ matrix with full column rank, why must $C^H B C$ be positive-definite?
In class, we analyzed a system with $n$ masses and $n+1$ springs and showed that it satisfied an equation $$ m \frac{d^2x}{dt^2} = -D^T K D x $$ for the vector $x \in \mathbb{R}^n$ of displacements, and we showed that $-D^T (K/m) D $ was negative-definite, and that this led to oscillating solutions.
Suppose that the masses are not identical, and let $M$ be the diagonal $n \times n$ matrix of masses $m_1,\ldots,m_n > 0$. If we define $y = \sqrt{M} x$, then show that $\frac{d^2y}{dt^2} = By$ where $B$ is negative-definite, and hence we still have oscillating solutions.
The nullspace $N(A)$ of the real matrix $A$ is spanned by the vector $v = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}$.
(a) Give an eigenvector and eigenvalue of the matrix $B = (3I - A^T A)(3I + A^T A)^{-1}$.
(b) Aside from the eigenvalue identified in (a), if you consider all other eigenvalues $\lambda$ of $B$, which of the following must be true? (Indicate all that apply.)
Justify your answer.
(c) Give a good approximate formula for $B^n \begin{pmatrix} 0 \\ -1 \\ 0 \\ 8 \end{pmatrix}$ for a large $n$ (give an explicit numerical vector).