I am struggling with the dimensions of matrices in the following problem. $J(y)=\sum_{k=1}^p (y(t)y^T(t-T))^2 \ \ \ \ \ \ \ \ \ \ where \ \ \ \ y(t)=W^Tx(t)\\ $
In this problem, the size of x(t) is NxM and W is Nx1 so y(t) is 1xM. I need to find the second derivative of J with respect to W. I know that the first derivative of J is: $2\sum_{k=1}^p [(y(t)y^T(t-T))][x(t)y^T(t-T)+x(t-T)y^T(t)] $
and based on my calculations the second derivative would be:
$2\sum_{k=1}^p [x(t)y^T(t-T)+x(t-T)y^T(t)][x(t)y^T(t-T)+x(t-T)y^T(t)] +(y(t)y^T(t-T)[x(t)x^T(t-T)+x(t-T)x^T(t)] $
my problem is that considering the dimensions of the matrices there is a mismatch between the dimensions of the first part of the second equation.To be more clear, the size of the matrices in the term $[x(t)y^T(t-T)+x(t-T)y^T(t)][x(t)y^T(t-T)+x(t-T)y^T(t)] $ is both Nx1 which results into not being able to calculate the multiplication. Is this the correct way to find the derivative of these equations?