Solution to Mathematics for Machine Learning Exercise 3.2

Consider $\mathbb R^2$ with $\langle ,\rangle$ defined for all $\mathbf x$ and $\mathbf y$ in $\mathbb R^2$ as $$\langle \mathbf x,\mathbf y\rangle :=\mathbf x^\top\mathbf A\mathbf y,\quad \mathbf A:=\begin{bmatrix}2 & 0\\ 1 & 2\end{bmatrix} .$$Is $\langle ,\rangle$ an inner product?

Solution: Let $\mathbf x=[x_1,x_2]^\top$, $\mathbf y=[y_1,y_2]^\top$. By direct computations, we have $$\langle \mathbf x,\mathbf y\rangle=2x_1y_1+x_2y_1+2x_2y_2$$ and $$\langle \mathbf y,\mathbf x\rangle=2x_1y_1+y_2x_1+2x_2y_2.$$Therefore, in general, we see that $\langle \mathbf x,\mathbf y\rangle \ne \langle \mathbf y,\mathbf x\rangle$. This implies that $\langle ,\rangle$ is not an inner product.