Given a matrix P_[nxn] which is symmetric and I want to integrate its product say A_[nxn]*P[nxn] using an integrator that takes in vectors only. I read up on vectorization process and landed on the Matrix differential book by Magnus and Neudecker which gives me this helpful formula
$\dot{P}_{3\times3} = A_{n\times n}\times P_{3\times3}$ after vectorization $\dot{P}_{9\times1} = [A_{n\times n}\bigotimes I_{3}]\times P_{9\times1}$
But in my case the P matrix is symmetric, so I don't need to integrate the lower triangle of the P matrix, so anyone can help me with a formula to evaluate the vectorization operator for the upper or lower triangle of the matrix? Instead of the kronecker product which vectorizes the whole matrix. This means that my P matrix will be reduced to 6 elements(in the upper triangle) and my transformed A matrix will be a 6x6 matrix instead of 9x9.
Edit: I want to point out that i want to force the matrix integrand to be symmetric.