Let us consider the following $2 \times 2$ matrix, $A$. $$ A = \begin{bmatrix} w_1^TP_{11}w_1 & w_1^TP_{12}w_2 \\ w_2^TP_{21}w_1 & w_2^TP_{22}w_2 \end{bmatrix} $$ where $P_{ij}$'s are $n\times n$ matrices and $w_j \in \mathbb{R}^n$'s are column vectors. Note that $w_i^TP_{ij}w_j \in \mathbb{R}$.
Let $W = [w_1, w_2]$. It seems that $A$ can be represented by $W$ along with another matrix involving $P_{ij}$. I am thinking to represent $A$ such that $W^TMW$ for some $M$. But it seems that it is impossible to present in that manner. I think the notion of the tensor product could help here, but not entirely sure.
Any answers/suggestions/comments will be very appreciated.
Let $\boldsymbol 0 \in \mathbb R^n$. Then\begin{align} A &= \begin{bmatrix} w_1^{\mathrm T} P_{11} w_1 & w_1^{\mathrm T} P_{12} w_2 \\ w_2^{\mathrm T} P_{21} w_1 & w_2^{\mathrm T} P_{22} w_2 \end{bmatrix} \\ &= \begin{bmatrix} w_1^{\mathrm T} \big[\begin{matrix} P_{11} w_1 & P_{12} w_2 \end{matrix}\big] \\ w_2^{\mathrm T} \big[\begin{matrix} P_{21} w_1 & P_{22} w_2 \end{matrix}\big] \end{bmatrix} \\ &= \begin{bmatrix} w_1^{\mathrm T} & \boldsymbol 0^{\mathrm T} \\ \boldsymbol 0^{\mathrm T} & w_2^{\mathrm T} \end{bmatrix} \begin{bmatrix} P_{11} w_1 & P_{12} w_2 \\ P_{21} w_1 & P_{22} w_2 \end{bmatrix} \\ &= \begin{bmatrix} w_1 & \boldsymbol 0 \\ \boldsymbol 0 & w_2 \end{bmatrix}^{\mathrm T} \begin{bmatrix} \begin{bmatrix} P_{11} \\ P_{21} \end{bmatrix} w_1 & \begin{bmatrix} P_{12} \\ P_{22} \end{bmatrix} w_2 \end{bmatrix} \\ &= \begin{bmatrix} w_1 & \boldsymbol 0 \\ \boldsymbol 0 & w_2 \end{bmatrix}^{\mathrm T} \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w_1 & \boldsymbol 0 \\ \boldsymbol 0 & w_2 \end{bmatrix} \\ &= W^{\mathrm T} M W. \end{align}
So that $$W = \begin{bmatrix} w_1 & \boldsymbol 0 \\ \boldsymbol 0 & w_2 \end{bmatrix} %= \big[\begin{matrix} w_1 & w_2 \end{matrix}\big] \begin{bmatrix} 1&0\\0&1 \end{bmatrix} ; \quad M = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix}.$$