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May I ask for the following form \begin{equation*} B = X^{T}AX \end{equation*} where $X$ is a matrix (the special case of size $n \times 1$ is excluded), what is the name of such forms? Quadratic matrix form? Since this form is also special, it should deserve a name.

Thank you!

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If $A$ is the matrix of a bilinear form on a based vector space $W$ and $X$ is the matrix of a linear mapping from another based vector space $V$ to $W,$ then $B$ is the matrix of the pullback of the form, which is a bilinear form on $V.$

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