Is there a standard way to approach solving bilinear expressions like below, where the parameter of interest, the only unknown, forms part of the matrix of the bilinear form
$$x^{T}Ay=\beta$$
where $A$ and $T$ are $n \times n$ diagonal matrix with elements $$A_{ii} = e^{\alpha T_{ii}},\: i \in \{1,...,n\} \text{ and } T_{ii}>0$$ and $x,y \in \mathbb{R}^{n}$ and $\beta > 0$ are given quantities.
Here, $-1<\alpha <1$ is the unknown parameter of interest and $x_{i},y_{i} >0 \: \forall i$.
One direct approach considered is to use $A = \exp{(\alpha T)}$, to obtain an approximate polynomial expansion in $\alpha T$ to some order. But is there a way to avoid expanding the exponential matrix to solve for $\alpha$?
You could select $x_{1} = e^{-\alpha T_{1,1}}$, $y_{1}=\beta$, and $x_{i} = y_{i} = 0, \forall i>1$, then $\alpha$ is any real number, or complex number if $x$ and $y$ accept complex values.