Criteria to find a common non orthonormal basis for two linear operators












0














I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check




  1. $A=A^+$

  2. $B=B^+$

  3. $[A, B] =0$


In this situation I can affirm that A and B have a common orthonormal basis of eigenvector, or if the 1 and 2 properties are not true for A and B but the third one is, I can say that A and B have only a common eigenvector.



But what properties should I check to know if two operators have a common non orthonormal basis? Can I ask this question to myself or it is incorrect itself?



Often on old tests I find "determine if this two operator have a common eigenvector basis. What kind of basis is this and why?" but I only find on my book criteria to find the orthonormal one.



Thank you and sorry for bad English.










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  • If they have a common basis of eigenvectors, then they satisfy 3. (it is enough to check this on the basis of common eigenvectors and there it is easy).
    – Severin Schraven
    Jan 4 at 16:55










  • For the other direction you might want to check this out mathoverflow.net/questions/124779/…
    – Severin Schraven
    Jan 4 at 16:58
















0














I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check




  1. $A=A^+$

  2. $B=B^+$

  3. $[A, B] =0$


In this situation I can affirm that A and B have a common orthonormal basis of eigenvector, or if the 1 and 2 properties are not true for A and B but the third one is, I can say that A and B have only a common eigenvector.



But what properties should I check to know if two operators have a common non orthonormal basis? Can I ask this question to myself or it is incorrect itself?



Often on old tests I find "determine if this two operator have a common eigenvector basis. What kind of basis is this and why?" but I only find on my book criteria to find the orthonormal one.



Thank you and sorry for bad English.










share|cite|improve this question
























  • If they have a common basis of eigenvectors, then they satisfy 3. (it is enough to check this on the basis of common eigenvectors and there it is easy).
    – Severin Schraven
    Jan 4 at 16:55










  • For the other direction you might want to check this out mathoverflow.net/questions/124779/…
    – Severin Schraven
    Jan 4 at 16:58














0












0








0







I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check




  1. $A=A^+$

  2. $B=B^+$

  3. $[A, B] =0$


In this situation I can affirm that A and B have a common orthonormal basis of eigenvector, or if the 1 and 2 properties are not true for A and B but the third one is, I can say that A and B have only a common eigenvector.



But what properties should I check to know if two operators have a common non orthonormal basis? Can I ask this question to myself or it is incorrect itself?



Often on old tests I find "determine if this two operator have a common eigenvector basis. What kind of basis is this and why?" but I only find on my book criteria to find the orthonormal one.



Thank you and sorry for bad English.










share|cite|improve this question















I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check




  1. $A=A^+$

  2. $B=B^+$

  3. $[A, B] =0$


In this situation I can affirm that A and B have a common orthonormal basis of eigenvector, or if the 1 and 2 properties are not true for A and B but the third one is, I can say that A and B have only a common eigenvector.



But what properties should I check to know if two operators have a common non orthonormal basis? Can I ask this question to myself or it is incorrect itself?



Often on old tests I find "determine if this two operator have a common eigenvector basis. What kind of basis is this and why?" but I only find on my book criteria to find the orthonormal one.



Thank you and sorry for bad English.







eigenvalues-eigenvectors operator-theory orthonormal normal-operator






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited Jan 4 at 17:09







pter26

















asked Jan 4 at 16:40









pter26pter26

317111




317111












  • If they have a common basis of eigenvectors, then they satisfy 3. (it is enough to check this on the basis of common eigenvectors and there it is easy).
    – Severin Schraven
    Jan 4 at 16:55










  • For the other direction you might want to check this out mathoverflow.net/questions/124779/…
    – Severin Schraven
    Jan 4 at 16:58


















  • If they have a common basis of eigenvectors, then they satisfy 3. (it is enough to check this on the basis of common eigenvectors and there it is easy).
    – Severin Schraven
    Jan 4 at 16:55










  • For the other direction you might want to check this out mathoverflow.net/questions/124779/…
    – Severin Schraven
    Jan 4 at 16:58
















If they have a common basis of eigenvectors, then they satisfy 3. (it is enough to check this on the basis of common eigenvectors and there it is easy).
– Severin Schraven
Jan 4 at 16:55




If they have a common basis of eigenvectors, then they satisfy 3. (it is enough to check this on the basis of common eigenvectors and there it is easy).
– Severin Schraven
Jan 4 at 16:55












For the other direction you might want to check this out mathoverflow.net/questions/124779/…
– Severin Schraven
Jan 4 at 16:58




For the other direction you might want to check this out mathoverflow.net/questions/124779/…
– Severin Schraven
Jan 4 at 16:58










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