can someone please explain self-adjoint operator intuition to me. And why when $T^* = T^{-1}$, $T$ preserves the inner product and therefore preserves the the orthonormal basis and the length and distance? thank you
2026-03-25 22:01:46.1774476106
self-adjoint operator intuition
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Think about a symmetric matrix, it’s a simpler case. I think that the term arise from an observation: matrices that are diagonal if we describe them with orthonormal bases have a symmetric structure if we change coordinates with an orthogonal transformation. This is a different way to interpret the def immediatley using the specrtal theorem.