(Note: in what follows, $B$ can be an arbitrary $n \times n$ matrix over $\mathbb{R}$ or $\mathbb{C}$, but $C$ must be an invertible matrix over the same field, for reasons I will discuss.)
If $C$ is orthogonal, and thus $C^T=C^{-1}$, I know that this coincides with a similarity transformation $B \mapsto P^{-1}BP$, which is the change of coordinates formula for a linear operator.
However, I am interested in the change of coordinates formula for quadratic forms (and in particular projective changes of coordinates for conic sections in $\mathbb{CP}^2$) and this is definitely only $B \mapsto C^T B C$ and not the same thing as matrix conjugation/similarity transformation.
Question: What is the name of the operation $B \mapsto C^TBC$?
I want to know the name of this operation so I can read up about it more.
What I know already: this type of transformation, unlike a similarity transformation, does not preserve the exact numerical value of the spectrum of $B$, but it preserves the number of zeroes in the spectrum (at least for symmetric and other diagonalizable matrices, but I suspect this holds in general due to the multiplicative rule for determinants). This fact can be used to show that the conic sections in $\mathbb{CP}^2$ form three distinct classes under projective changes of coordinates. Also, the fact that bilinear forms transform this way under changes of coordinates, instead of via similarity transformations like linear operators do, shows that they are a different type of second-order tensor compared to linear operators.
Anyway, I would like to know a specific way to refer to this transformation so I can look it up in the literature and feel more confident about my ability to tell it apart it from similarity transformations.
The standard terminology for a change of basis for the matrix representation of a quadratic form is matrix congruence.