Consider the vectors $\mathbf{a} \in \mathbb{R}^N$ and $\mathbf{b} \in \mathbb{R}^N$ with $N>1$ and $\mathbf{a} \neq \mathbf{b}$. The product $\mathbf{C}=\mathbf{a} \mathbf{b}^T \in \mathbb{R}^{N \times N}$ will yield a square, non-symmetric matrix $\mathbf{C}$ with rank=1.
How do I formulate an optimization problem on $\mathbf{a}$, given a specific $\mathbf{b}$, such that $\mathbf{C} = \mathbf{a} \mathbf{b}^T$ is negative definite? How do I specify this constraint?
$$ \min_{\mathbf{a}} \text{a user-specified objective} \\ \text{subject to} \ldots ?$$
I know, a square, nonsymmetric matrix $\mathbf{D}$ is negative definite if and only if its symmetric part $(\mathbf{D} + \mathbf{D}^T)/2$ is negative definite. However, in my case, the symmetric part $(\mathbf{C} + \mathbf{C}^T)/2$ will have a maximum rank of 2, and it may have zero eigenvalues... See also this wiki-link.
