I often encounter things of the form $x^\top M x$, where $M$ is symmetric positive (semi-)definite. Is there a term for that? I know related terms:


* We can say $M$ is a [bilinear form](https://en.wikipedia.org/wiki/Bilinear_form), $M(x,y) = x^\top M y$ with an induced norm, $|\!|x|\!|\_M := \sqrt{x^\top M x}$, so I could call $x^\top M x = \!\|x|\!|\_M^2$ "the squared norm of $x$ with respect to $M$ I guess?
* This is related to [Mahalanobis distance](https://en.wikipedia.org/wiki/Mahalanobis_distance), although that's very particular to statistics.
* This is related to the idea of a [metric tensor](https://en.wikipedia.org/wiki/Metric_tensor) where $M$ is the metric tensor and .
* It looks like this can be called a ["change of basis" of a bilinear form](https://en.wikipedia.org/wiki/Change_of_basis#The_matrix_of_a_bilinear_form).


So: In general, is there a name for $f(M, x) = x^\top M x$? There must be, right?



Is there a name for $f(M, x) = x^\top M x$?
You can say that $x^\top M x$ is the quadratic form associated to $M$.


