Consider the set $P_n$ of all polynomials of degree $\le n$ with complex coefficients. Any such polynomial $p$ can be represented as a coefficient vector $[p]\in \Bbb C^{n+1}$ via $$p(x) = \sum_{i=0}^n a_ix^i \Longleftrightarrow [p] = \left[\begin{matrix}a_0 \\a_1\\ \vdots \\a_n \end{matrix}\right]$$ Suppose we define an inner product on polynomials as $(p,q) = \int_{-1}^1 \overline{p(x)} q(x)dx$. Thus, two polynomials $p,q$ are orthogonal if $\int_{-1}^1 \overline{p(x)} q(x)dx = 0$.
(a) Show that there exists a Hermitian matrix $G$ such that $(p,q) = [p]^*G[q]$ for all polynomials $p,q\in P_n$.
I know that that this is the polynomial analogue of QR factorization, but I cannot see how to get started. I thought of proving the general definition by somehow but I don't understand how to deal with the LHS that is $(p,q)$.
You can write down the inner product explicitly. What is $(x^n, x^m)?$ It will be the $n,m$ entry of your mysterious $G.$