Question on the adjoint of a function

Question

• 1) Let $E$ a vector space of finite dimension with scalar product $\left<\cdot,\cdot\right>$. We defined the adjoint of $f:E\to E$ the function $g$ defined as $$\left<f(x),y\right>=\left<x,g(y)\right>.$$

Let $S$ s.t. $\left<x,y\right>=x^TSy$. Let $F$ the matrix of $f$ and $G$ the matrix of $g$. If $S=I$, then $$\left<Fx,y\right>=x^TF^Ty=\left<x,F^Ty\right>$$ and thus $G=F^T$. But if $S\neq I$, we have $$\left<Fx,y\right>=x^T F^T S y,$$ and how to get $G$ s.t. $\left<x,Gy\right>=\left<Fx,y\right>$ ?

• 2) We define in a more general way, the adjoint of $f\in L(E,F)$ as $g\in L(F^*,E^*)$ s.t. $g=f\circ u$. What is the link between the adjoint defined in 1) and the adjoint defined in 2) ?

Answer 1

A change of basis is not necessary, just take $G = S^{-1} F^T S$: $$\langle Fx,y\rangle = x^TF^TSy = x^TSS^{-1}F^TSy = \langle x,S^{-1}F^TSy\rangle.$$

Answer 2

1) There is a basis s.t. $S$ is congruent to $I$ (i.e. there is a matrix $P$ invertible s.t. $S=P^TP$. Let denote $A=P^{-1}FP$ the matrix of $f$ in the new basis.

Then $$\langle APx,Py\rangle= x^TP^TA^TPy=\langle Px,A^TPy\rangle,$$ and thus $F^T$ is the matrix of $g$ in the new basis. $$F=PAP^{-1}\quad \text{and}\quad G=PA^TP^{-1}.$$

2) You have that $$\left\langle u(x),f\right\rangle_{F,F^*}=\left\langle x,u^T(f)\right\rangle_{E,E^*}.$$