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If we assume that $D$ is nonsingular, then there is a relatively straightforward (and efficient) solution based on an $LU$ decomposition. If we write $$ \pmatrix{D & B \\ B^T & A} = \pmatrix{ L_{11} & \\ L_{21} & L_{22}} \pmatrix{U_{11} & U_{12} \\ & U_{22}} = \pmatrix{L_{11} U_{11} & L_{11} U_{12} \\ L_{21} U_{11} & L_{21} U_{...


At least for the second question the answer is yes. See for example Mattheij, Robert MM, and Gustaf Söderlind. "On inhomogeneous eigenvalue problems. I." Linear Algebra and its Applications 88 (1987): 507-531, page 516. (The optimality conditions of your problem, $$ Ax+b+2\lambda x = 0 \\ x^Tx = 1 $$ constitute an inhomogenous eigenvalue problem)


Unfortunately, I don't think there is a good algorithm to do this efficiently. Given the eigendecomposition $\mathbf A = \mathbf X \mathbf D \mathbf X^T$, one is tempted to project $\mathbf v$ onto the eigenvectors by introducing the vector $\mathbf u = \mathbf X^T \mathbf v$, forming $\mathbf A + \mathbf v \mathbf v^T = \mathbf X \left(\mathbf D + \mathbf u ...

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