$A$ is a symmetric $n\times n$ matrix with strictly positive eigenvalues.
Define $( \cdot ,\cdot) : \mathbb{R}^{ n} \times \mathbb{R}^{ n} $ $\rightarrow \mathbb{R}$ by $(x,y) = x' A y$.
Given that $T$ is an arbitary linear operator $T: \mathbb{R}^{ n} \times \mathbb{R}^{ n} $
Could please show me analytically how would you find the adjoint of $T$?
Assuming $'=^\text{T}$ denotes transposition. The hypothesis on $A$ assure that $(\cdot,\cdot)$ is an inner product and $A$ is invertible.
By definition, $T^\ast$ is defined by : $$(x,Ty)=(T^\ast x,y).$$ We have : $$(x,Ty)=x'ATy=x'ATA^{-1}Ay=[A^{-1}T^\text{T}Ax]'Ay=(A^{-1}T^\text{T}Ax,y),$$ which means : $$T^\ast=A^{-1}T^\text{T}A.$$