Criteria to find a common non orthonormal basis for two linear operators
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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 $A=A^+$ $B=B^+$ $[A, B] =0$ 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 sor
