Suppose you have an operator ## T: V → V ## on a finite-dimensional inner product space, and suppose it is orthogonally diagonalizable. Then there exists an orthonormal eigenbasis for V. Is this eigenbasis unique?(adsbygoogle = window.adsbygoogle || []).push({});

Obviously, in the case of simple diagonalization, the basis is not unique since scaling (by nonzero) any vector in an eigenbasis yields a valid eigenbasis.

Likewise, an orthonormal basis for a space of at least dimension 2 is not unique, since we can take any two nonparallel vectors in the space and extend each to its own orthonormal basis through Gram-Schmidt. The two bases must be distinct.

But what about an orthonormal eigenbasis? Is this set unique? My guess is that it is, but I need to know for sure so I can think about which direction I want to steer my proof.

Thanks!

BiP

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# Are orthonormal eigenbases unique?

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