Decomposing a matrix into two or more independent matrices

Question

I am almost certain the solution to the problem is very obvious but I can't see an easy way of doing it.

I would like to implement a solution based on matrix multiplication to compute the following expression: $$ H = \sum_{i,k}M^2_{i,k} + \sum_{i,k}\sum_{(j,l) \neq (i,k)}\phi_{i,j}\,\rho_{k,l}\,M_{k,i}\,M_{l,j} $$

For the sake of simplicity, I have bounded the variables $i$ and $k$ to be $i = 0,1$ and $k = 0,1$. I can now write the above expression as a matrix multiplication problem in the following way: $$ \left( \begin{array}{cccc} M_{00} & M_{01} & M_{10} & M_{11}\end{array} \right) \times C \times \left( \begin{array}{} M_{00} \\ M_{01} \\ M_{10} \\ M_{11} \end{array} \right) $$ where, if I havent made any mistakes, $C$ would be defined as $$ C = \left( \begin{array}{cccc} 1 & \phi_{01}\,\rho_{00} & \phi_{00}\,\rho_{01} & \phi_{01}\,\rho_{01}\\ \phi_{10}\,\rho_{00} & 1 & \phi_{10}\,\rho_{01} & \phi_{11}\,\rho_{01}\\ \phi_{00}\,\rho_{10} & \phi_{01}\,\rho_{10} & 1 & \phi_{01}\,\rho_{11}\\ \phi_{10}\,\rho_{10} & \phi_{11}\,\rho_{10} & \phi_{10}\,\rho_{11} & 1\end{array} \right) $$ My question is:

How can I express the matrix $C$ as a multiplication of 2 or more matrixes such that the coefficients for $\phi$ and $\rho$ are in different matrixes? Basically I would like to have a simple way of storing this data in different matrixes, and then just simply multiply all my matrixes to solve the combinations that give me $H$.

Answer 1

Ignore the summation constraints for a moment, and write the function as $$\eqalign{ H &= M:M + M:\rho M\phi^T \cr &= {\rm vec}(M):{\rm vec}(M) + {\rm vec}(M):{\rm vec}(\rho M\phi^T) \cr &= {\rm vec}(M)^T{\rm vec}(M) + {\rm vec}(M)^T(\phi\otimes\rho){\rm vec}(M) \cr &= m^Tm + m^TGm\cr }$$ where : represents the inner/Frobenius product, $\otimes$ represents the Kronecker product, and ${\rm vec}()$ is the matrix vectorization operator.

The constraint says something about the intrinsic structure of the $G$ matrix.

From your sample calculation, it appears the constrained matrix has its diagonal elements removed. This can be expressed in a variety of ways $$\eqalign{ G^\prime &= G - I\odot G \cr &= G - {\rm Diag}(G) \cr &= G - D \cr }$$ where $\odot$ represents the elementwise/Hadamard product, and $I$ is the identity matrix.

Substituting the constained matrix yields $$\eqalign{ H &= m^Tm + m^TG^\prime m \cr &= m^TIm + m^TGm - m^TDm \cr &= m^T\,(I+G-D)\,m \cr &= m^T\,C\,m \cr\cr }$$ You were mainly interested in an expression for the $C$ matrix $$\eqalign{ C &= I \,+ (J-I)\odot(\phi\otimes\rho) \cr }$$ where $J$ is the all ones matrix, which is the identity element for the Hadamard product.