Show that $\langle x,y \rangle_A$=$(x_1,x_2...x_n)A(y_1,y_2,...y_n)^T$

Question

Let $A$ be a $n \times n$ matrix and let and let $x,y \in R^n$ with $x=(x_1,x_2...x_n)^t$ and $y=(y_1,y_2,...y_n)^t$ then we define $\langle x,y \rangle_A$=$x \cdot (Ay)$ with $x \cdot y$ the standard dot product. Show that $\langle x,y \rangle_A$=$(x_1,x_2...x_n)A(y_1,y_2,...y_n)^T$.

I only know how I can prove this when $A$ is symmetric. But it's not given that $A$ is symmetric so I can't use that.

I was thinkin about using $\langle x,y \rangle_A$=$x^T(Ay)$=$(x^TA^T)y$=$(Ax)^Ty$ and so on, but I can't since $A \neq A^T$. It does say that it's a standard dot product, and $\langle x,y \rangle_A$ only is a dot product if $A$ is symmetric, but I'm not sure if I can assume that $A$ always is symmetric.

Answer 1

You essentially already have it: $\langle x,y \rangle_A=x^T(Ay)=x^TAy$ (since matrix multiplication is associative, we can remove parentheses) =$(x_1,x_2...x_n)A(y_1,y_2,...y_n)^T$ by writing out the entries of $x,y$.