Basic operation geometrical meaning [closed]

Question

What is the geometrical meaning of doing $x^TAx \;$?

$Ax \; $ is trivially "applying A to x", but then, what the multiplication for $x^T$ stands for?

Answer 1

In $x^TAx$, you should not think of $A$ as a linear operator; you should think of $A$ as defining a bilinear form or quadratic. If you take $A=I$, the identity matrix, then $x^Tx$ is the dot product of the vector $x$ with itself and $x^Ty$ is the dot product of the vectors $x$ and $y$; taking a different $A$ is replacing the dot product by a new bilinear form in which the product of the $i$th and $j$th standard basis vectors is $a_{ij}$.