Determine a matrix given eigenvalues and single eigenvector? Anyway, say there is some original matrix $M \in \mathbb{R}^{n\times n}$ that you are trying to reconstruct.  Also let $M$ be symmetric and positive-definite.  You have at your disposal $\Lambda$, a diagonal matrix who contains its eigenvalues and only a single normalized eigenvector, a column vector $v_1 \in \mathbb{R}^{n\times1}$.  Originally I thought it would be possible because the eigenvectors of a symmetric matrix are orthogonal, so we could just find a basis for the null space of $v_1$.
For example do an singular value decomposition of $v_1$ so $v_1=USV$ where obviously $V=1$, as $v_1^Tv_1=1$ since $v_1$ is normalized.  Also $v_1$ can only have rank $1$, so $S=[1,0,0,\ldots,0]^T$.  Lastly we consider $U \in \mathbb{R}^{n\times n}$, found from eigenvectors of $v_1v_1^T$, which should be orthogonal as $v_1v_1^T$ is also symmetric. Breaking $U$ into column vectors gives $U=[u_1,u_2,\ldots,u_n]$.  See $v_1=u_1$ as
$$v_1 = [u_1,u_2,\ldots,u_n] \cdot [1,0,0,\ldots,0]^T\cdot1=u_1.$$
Thus, since $u_2,\ldots,u_n$ are orthogonal to $u_1$, and therefore $v_1$, we have a set of orthogonal vectors we could potentially use to reconstruct $M$ by its diagonalization, so
$$M=U\cdot\Lambda\cdot U^T.$$
Thoughts on why this process is wrong?

"Given $\Lambda$ and $v_1$, construct $M$" is your question in short. 
Let $\lambda_k$ and $v_k$ be the $k^{th}$ eigenpair. $M$ is written as $$M = \sum\limits_{k=1}^N \lambda_k v_kv_k^T$$
In the above equation, you have $\lambda_1 \ldots \lambda_N$ and $v_1$; for complete reconstruction, you need $v_2 \ldots v_N$.
