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May 18, 2023 at 11:27 comment added Igor Khavkine This old answer by Peter Michor links to a survey of results that could help if you relax the premise of your question. Namely, you could potentially find a continuous mapping $V\to U$, which "blows up" the points where $G(p)$ is singular such that the diagonalization of $G$ is continuous over $V$.
May 17, 2023 at 5:21 vote accept user1234567890
May 16, 2023 at 17:38 comment added user1234567890 @ChristianRemling no it's not. That's the case for matrices over a field, but not over a general ring, and that's what I'm asking.
May 16, 2023 at 16:50 comment added Christian Remling Your matrix is symmetric, so is diagonalizable everywhere. (A different question that is more interesting was answered by @RobertBryant below.)
May 16, 2023 at 16:15 history became hot network question
May 16, 2023 at 12:31 answer added Robert Bryant timeline score: 8
May 16, 2023 at 7:00 history asked user1234567890 CC BY-SA 4.0