- #1
Bipolarity
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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?
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
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