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I am just curious about how to use a choice of basis for general V, to decide whether L is self-adjoint. The issue, specifically, is that the relation ## L= L^T ## ( abusing notation ; here L is a matrix representing L in some choice of basis ) which holds for self-adjoint operators in f.dim. space, will most likely not hold under a change of basis. But we may be able to find a special basis in a given V:

I think for ## \mathbb R^n ## , if we use the standard basis e_i=δ

_{i,j}, then L is

self adjoint if , when it is represented as a matrix M in this basis, we have that ## M^T = M ## , i.e., M equals its transpose ( if V is complex, we need the resp. matrix to equal the transpose of the conjugate ) . (Phew !) Now, can we find some specific basis ## B_V ## in a general f.dim vector space V so that we can conclude L : V-->V is self adjoint if/when its representing matrix M satisfies ## M= M^T ## (or equals its conjugate transpose if the base field is C)? I thought we may use an vector space isomorphism between V and ## \mathbb R^n ## to pull back the basis {#e_i#} , and then this "pulled-back" basis would do the job?

Thanks in Advance.