Convert between matrix system of equations

Question

I have a system of equations defined by:

$\phi^T M(\alpha) \phi$

That is, some vectors with scalar entries ($\phi$) and a matrix $M$ parameterised by $\alpha$ in some way. I would like convert this equation into the form:

$A\mathbf{\alpha}$ where $\mathbf{\alpha} = [\alpha_1 , \alpha_2 ...,\alpha_m]^T$ and $A$ is a matrix of some scalar coefficients. This is the standard form for a linear system of equations.

I was wondering how one would approach this re-arrangement? Is it really just about trying to expand one big example matrix and keeping track of patterns, or is there some mathematical way to prove the form/manner in which this re-arragement would take place?

Thanks!

Answer 1

$\phi^TM(\alpha)\phi$ is a single term, i.e. if $\phi \in \mathbb{R}^{n \times 1}$ and $M(\alpha) \in \mathbb{R}^{n \times n}$, we have $\phi^TM(\alpha)\phi \in \mathbb{R}^1,$

However, $A\alpha$ is a vector. If $A \in \mathbb{R}^{m \times m}$ and $\alpha \in \mathbb{R}^{m}$, then $A\alpha \in \mathbb{R}^m$.

Hence, such conversion is impossible.

Remark:

If your intention is to ask about $\alpha^TA\alpha$ then expression of $M(\alpha)$ should be an expression up to quadratic terms only.