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

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In summary, the adjoint of a linear map is not always guaranteed to be equivalent to the conjugate transpose of the matrix, especially when the basis is not orthonormal. In this case, the basis determines whether the matrix of the adjoint can be considered as the conjugate transpose of the matrix of the map. However, the orthonormality of the basis depends on the inner product being used.
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balletomane
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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)
 
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  • #2
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.
 
  • #3
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.
 

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

1. What is the difference between an adjoint and a conjugate transpose?

An adjoint is the Hermitian transpose of a matrix, which is the transpose of the complex conjugate of the matrix. A conjugate transpose is the transpose of the complex conjugate of a matrix. In other words, the adjoint is the conjugate transpose of a matrix.

2. How do you calculate the adjoint of a matrix?

To calculate the adjoint of a matrix, you need to find the Hermitian transpose of the matrix. This can be done by taking the transpose of the complex conjugate of the matrix, which involves changing the signs of the imaginary components of the matrix entries. The resulting matrix is the adjoint of the original matrix.

3. What is the purpose of the adjoint and conjugate transpose in linear algebra?

The adjoint and conjugate transpose are used in linear algebra to solve systems of equations and to calculate inner products of vectors. They are also important in the study of operators and transformations in vector spaces.

4. Can the adjoint and conjugate transpose be applied to non-square matrices?

Yes, the adjoint and conjugate transpose can be applied to non-square matrices. However, the resulting matrix will not be a square matrix and will have different dimensions than the original matrix.

5. Are there any properties or rules for adjoints and conjugate transposes?

Yes, there are several properties and rules that apply to adjoints and conjugate transposes, including the fact that the adjoint of a product of matrices is equal to the product of the adjoints in reverse order, and that the adjoint of the inverse of a matrix is equal to the inverse of the adjoint. Additionally, the adjoint and conjugate transpose are both linear operators, meaning they follow the properties of linearity such as the distributive and associative properties.

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