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I learned in quantum mechanics that the eigenvectors of a Hermitian operator (matrix) can be made to form an orthogonal eigenbasis for the vector space.

I also learned that 2 commuting Hermitian operators share a common eigenbasis. Does this mean that Hermitian operators can have more than one eigenbasis? In general, how many eigenbases can a Hermitian operator have?

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arbitrarily many, just take the matrix corresponding to the scalar multiplicattion, this has every basis as an eigenbasis

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  • $\begingroup$ do you mean if ${ \vec{x}, \vec{y}, \vec{z} }$ form an eigenbasis for a Hermitian operator, ${ A \vec{x}, A \vec{y}, A \vec{z} }$ is another one, where A is a scalar? $\endgroup$ – TaeNyFan Feb 15 at 15:35
  • $\begingroup$ is it possible to have $\vec u, \vec v, \vec i$ as another eigenbasis, where these are not necessarily ${ A \vec{x}, A \vec{y}, A \vec{z} }$? $\endgroup$ – TaeNyFan Feb 15 at 15:37
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It is not true that two commuting Hermitian operators share a common eigenbasis, as you can see in this question, but if two operators share an eigenbasis, that is, they are simultaneously diagonalizable, they must commute.

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If the eigenvalues of a diagonalizable operator $A$ all have multiplicity $1$ then the eigenbasis is unique up to multiplication of the eigenvalues by scalars. There is essentially just one basis of eigenvectors.

Any operator commuting with $A$ will have the same eigenvectors as $A$. To see why, suppose $v$ is an eigenvector for $A$. Then $$ ABv = BAv = B\lambda v = \lambda Bv $$ so $Bv$ is a $\lambda$-eigenvector of $A$. Since the space of those eigenvectors is one dimensional and spanned by $v$, $Bv$ must be a multiple of $v$, hence an eigenvector of $B$ for some eigenvalue.

I suspect that the multiplicity $1$ assumption is implicit in the quantum mechanical application you have in mind.

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The identity $I$ has a lot eigen bases. Any time an eigenspace has dimension $> 1$, there are different eigen bases. If the multiplicities of the eigen bases are all $1$, then your basis elements are unique up to unimodular constant multipliers.

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