Summary: We discuss the eigenproblem for the symmetric arrowhead matrix \(C=\begin{pmatrix} D & \mathbf{z}\\ \mathbf{z}^T & \alpha\end{pmatrix}\), where \(D\in \mathbb{R}^{n\times n}\) is diagonal, \(\mathbf{z}\in\mathbb{R}^n\), and \(\alpha\in\mathbb{R}\), in order to examine criteria for when components of \(\mathbf{z}\) may be set to zero. We show that whenever two eigenvalues of \(C\) are sufficiently close, some component of \(\mathbf{z}\) may be deflated to zero, without significantly perturbing the eigenvalues of \(C\), by either substituting zero for that component or performing a Givens rotation on each side of \(C\). The strategy for this deflation requires \(\mathcal{O}(n^2)\) comparisons. Although it is too costly for many applications, when we use it as a benchmark, we can analyze the effectiveness of \(\mathcal{O}(n)\) heuristics that are more practical approaches to deflation. We show that one such \(\mathcal{O}(n)\) heuristic finds all sets of three or more nearby eigenvalues, misses sets of two or more nearby eigenvalues under limited circumstances, and produces a reduced matrix whose eigenvalues are distinct in double the working precision. Using the \(\mathcal{O}(n)\) heuristic, we develop a more aggressive method for finding converged eigenvalues in the symmetric Lanczos algorithm. It is shown that except for pathological exceptions, the \(\mathcal{O}(n)\) heuristic finds nearly as much deflation as the \(\mathcal{O}(n^2)\) algorithm that reduces an arrowhead matrix to one that cannot be deflated further. The deflation algorithms and their analysis are extended to the symmetric diagonal-plus-rank-one eigenvalue problem and lead to a better deflation strategy for the LAPACK routine dstedc.f.