I've seen the term $x^T A x$ come up in a bunch of different areas of linear algebra, where A is a square and usually symmetric matrix. Places I've seen it include defining the Raleigh quotient, defining positive/negative semi-definite matrices, and in the derivation of PCA. I've also seen it sometimes referred to as describing a quadratic form.
Is there some general definition/ intuitive description of what $x^T A x$ means with respect to a vector and a matrix? My sort of vague understanding is that it describes how a vector is changed under a linear transformation defined by A (for example if A causes x to rotate 90 $^\circ$ then $x^T A x = 0$) but I can't seem to come up with a more precise or insightful description of $x^T A x$, and I'm surprised how little I could find online considering how often I see this term come up.