Given a symmetric $m \times m$ matrix $X$, and say we know that the smallest eigenvalue of $X$ has eigenvector $u$. Is there a way to use this information to construct a $(m-1)\times (m-1)$ matrix $Y$, such that $Y$ has the same first $m-1$ eigenvalues as $X$?
2026-03-30 15:08:47.1774883327
Eigenvalue preserving projection
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Hint: Rotate $X$ to a coordinate system where $u$ is parallel to one of the axes. Since the eigenvectors of $X$ must be orthogonal, what does this new matrix look like? Can you find $Y$ from there?