For a bilinear functional why we have $b(x,Ay) = b(A^Tx,y)$?

Question

For a bilinear functional, or an inner product, why we have $b(x,Ay) = b(A^Tx,y)$, or $(x,Ay)_\Omega = (A^Tx,y)_\Omega$, for a matrix $A$?

Answer 1

As Rahul pointed out in the comments what you have written is in general false. But if we restrict ourselfs to $b$ being the euclidean inner product on $\mathbb{R}^n$ it becomes correct. You just have to write the inner product as matrix multiplications: $$(x,Ay)=x^T(Ay)=x^TAy=(A^Tx)^Ty=(A^Tx,y).$$

If you talk about a general $b$ and a linear continuous map $A$ you can define the adjoint of $A$ (denoted with $A^*$) with respect to $b$ by having to satisfy $$b(x,Ay)=b(A^*x,y).$$ Hence if $b$ is the euclidean inner product as above you have $A^*=A^T$. See https://en.wikipedia.org/wiki/Hermitian_adjoint for more details