Equivalence of Adjoint and Conjugate Transpose in Non-Orthogonal Bases?

[balletomane]

Is the adjoint of linear map only guaranteed to be equivalent to the conjugate transpose of the matrix when the matrix is taken with respect to an orthonormal basis? Is it sometimes still equivalent even when the basis is not orthonormal?

For the problem I'm working on, I have T(a+bx+cx^2)=bx, which is not self adjoint. Then the matrix wrt to the basis (1,x, x^2) is zero everywhere, but 1 in the second row, second column. The only explanation I can come up with for why it's matrix equals the conj. transpose of the matrix even though it is not self adjoint is that the basis is not orthonormal.

(sorry I don't know how to format things properly)

 

[matt grime]

Since every basis is, with respect to some inner product, orthonomal, I don't think this is the issue...

You say 'the linear map' and the matrix'. Linear maps and matrices are different things. A matrix is a representation of a linear map with respect to some basis. It does not make sense to talk of a linear map being conjugate to a matrix. Matrices are conjugate to matrices.

 

[balletomane]

Sorry, I was being sloppy. I meant the matrix of the linear map. I guess I'm still a little unclear on whether the basis determines if you can consider the matrix of the adjoint as the conjugate transpose of the matrix of the map .

Either way, I hadn't noticed the point about the orthonormality depending on the inner product. So thanks.