Is this expression a quadratic form I have an matrix expression that basically is of the form:
\begin{equation}
tr(B X BX )
\end{equation}
Where $B$ and $X$ and nonsquare matrices.  $B$ is $p \times n$, $X$ is $n \times p$.  
It seems to me this trace expression is a quadratic form because I got it as part of a longer matrix expression which represents the Hessian term of a Taylor series approximating a matrix function.  However, I can't see to get it into the form:
\begin{equation}
\text{vec}(X)^T [H] \text{vec}(X)
\end{equation}
Where $[H]$ is the square matrix representation of this quadratic form.  I want to get it into this form so I can get the eigenvectors of this matrix $[H]$.  $\text{vec}(X)$ is the vector of length $pn$ where the columns of $X$ are on top of each other. 
Can someone help me confirm whether this trace expression can indeed be converted into an explicit matrix representation?
I know that I can make this into:
\begin{equation}
\text{vec}(X^T)^T (B^T \otimes B) \text{vec}(X)
\end{equation}
But this is not what I want because it gives me $\text{vec}(X^T)$ on the left!  It seems so close yet so far.
Thanks.

Just think of the matrix $U = XB.$ Your matrix is $U^2,$ so write down the expression for the trace of the square, then substitute, then find the coefficient of $x_{ij}.$ It's tedious, but not difficult.
