The following came up as part of another problem I was looking at involving quadratic forms. Given a real symmetric 3x3 matrix $\mathbf{A}$ with $|\mathbf{A}| = 0$, find two vectors $\mathbf{u}$ and $\mathbf{v}$ such that $\mathbf{A} = \mathbf{u}\mathbf{v}^T + \mathbf{v}\mathbf{u}^T$.
Is this always solvable? If so, is there a simple way to represent $\mathbf{u}$ and $\mathbf{v}$ in terms of $\mathbf{A}$?
Let A be a diagonal matrix with two of the diagonal entries equal to 1, and the third one equal to 0. For such a matrix A, there do not exist vectors u,v with real coefficients for which the specified relation holds.