Due Wednesday, November 28 at 10:55am.
Suppose that $A$ is a real $3\times 2$ matrix.
(a) If you solve the ODE $\frac{dx}{dt} = -A^T A x$ for some initial condition $x(0)$, then which of the following are possible behaviors for $x(t)$?
(b) For each of the possible behaviors you indicated in part (a), give a matrix $A$ and a corresponding initial condition $x(0)$ which will lead to that behavior.
(c) For each of the matrices you gave in part (b), compute the singular value decomposition: give the singular values $\sigma$ and the corresponding left and right singular vectors.
In class and in the textbook, positive-definiteness was defined only for Hermitian matrices. However, it can be extended to arbitrary square matrices as follows:
Show that if $A + A^H$ is positive definite, then the eigenvalues of $A$ have positive real parts.
(Hint: consider an eigenvector $Ax = \lambda x$. How can you use this in one of the positive-definite conditions for $A+A^H$ above?)