It seems I just cannot stop talking about widely linear equations. Sorry for being to monotonous. My colleague Dominik recently made me aware of the fact that it is possible to solve the widely linear system of equations $$ \ma{A} \cdot \ma{x} + \ma{B} \cdot \ma{x}^* = \ma{c}$$ that I already visited in an earlier […]

# Tag Archives: Algebra

## Widely Linear Least Squares

Together with my colleague Jens I recently stumbled across a widely linear estimation problem. More precisely, we were trying to estimate a vector $\ma{x} \in \compl^{N}$ such that $\ma{C} \cdot \begin{bmatrix}\ma{x}\\ \ma{x}^*\end{bmatrix}$ was as close as possible to a given $\ma{b} \in \compl^{M}$ for a given matrix $\ma{C} \in \compl^{M \times 2N}$. This is not […]

## Widely linear systems of equations

I would strongly assume that this must exist already somewhere but I couldn’t find the solution so I thought it would be interesting to post it here. The closest to this I could find is widely linear estimation (e.g., Picinbono, TSP, 1995), but it’s not quite the same. Consider the following widely linear system of […]

## … and a simpler proof for it

A much simpler proof for the more generic algebraic rule on manipulating quadratic forms I posted recently just became apparent to me. All you need to do is to first show that $${\rm trace}\{\ma{A}^{\rm H} \cdot \ma{B}\} = {\rm vec}\{\ma{A}\}^{\rm H} \cdot {\rm vec}\{\ma{B}\},$$ which is fairly easy because both represent a short-hand notation for […]

## Even more algebra fun

I just realized that there is an even more general version of the rules for quadratic forms I posted recently: $${\rm trace}\left(\ma{A} \cdot \ma{X} \cdot \ma{R} \cdot \ma{X}^{\rm H} \cdot \ma{B}^{\rm H}\right) = {\rm vec}\left(\ma{X}\right)^{\rm H} \cdot \left( \ma{R}^{\rm T} \otimes \ma{B}^{\rm H} \cdot \ma{A} \right) \cdot {\rm vec}\left(\ma{X}\right) $$ Enjoy!