Conjugate of a matrix and of a function

[DeepSeeded]

Hello,

Working without complex numbers a conjugate of any function in a LVS is always the same thing. A conjugate of any matrix in a LVS is very often not the same thing though. I am just confused as to why functional spaces rely on complex numbers for the conjugate to have any importance and a matrix does not.

 

[HallsofIvy]

Where did you see that? The "conjugate" of a matrix is just the matrix with the entries replaced by there complex conjugates. If M is a matrix with all real entries then the conjugate of M is just M itself.

You may be confusing "conjugate" with the "conjugate transpose" or "Hermitian transpose" of a matrix: swap rows and columns and take the conjugate of each entry. Of course, if M has all real entries, it "conjugate transpose" is just its transpose.

 

[DeepSeeded]

Where did you see that? The "conjugate" of a matrix is just the matrix with the entries replaced by there complex conjugates. If M is a matrix with all real entries then the conjugate of M is just M itself.

You may be confusing "conjugate" with the "conjugate transpose" or "Hermitian transpose" of a matrix: swap rows and columns and take the conjugate of each entry. Of course, if M has all real entries, it "conjugate transpose" is just its transpose.

So guess my question is if functions are a different represenation of a matrix why is there no option to transpose a function?

 

[HallsofIvy]

In what sense is a function a "different representation of a matrix"? Are you talking about representing linear functions represented by a matrix?

 

[DeepSeeded]

In my QM class Operator functions are said to be like a matrix.

 

[Landau]

I don't fully understand your question, but maybe you'd like to hear about the adjoint of a linear transformation.

Let $$V,W$$ be inner-product spaces, let $$T\in L(V,W)$$ be a linear transformation, and $$T^*\in L(W,V)$$ its adjoint. This means that $$\langle Tv,w \rangle=\langle v,T^*w \rangle$$ for all $$v\in V,w\in W$$. Then, the matrix of $$T^*$$ with respect to orthonormal bases of $$V$$ and $$W$$ is just the conjugate transpose of the matrix of $$T$$ with respect to these bases. As mentioned earlier, the conjugate transpose of a matrix is just the transpose (interchange rows and colums) of the matrix with all entries replaced by their complex conjugates.