Notation on inner product?

Question

If we are in normed spaces say $\mathbb{R}^n$, $\langle y,x\rangle = y^Tx$ where $y^T$ is transpose of $y$. But if $A$ is an $n \times n$ matrix, then why does $\langle Ax,Ax\rangle = x^TAx$? Shouldnt it be $\langle Ax,Ax\rangle = x^TA^TAx$?

Answer 1

Of course it should be $x^T A^T A x$. The only case where it would be $x^T A x$ would be if $A^T A = A$ (in particular if $A$ is an orthogonal projection).