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!

Fun with statistics – transformations of random variables part 1

Here is another tool I learned about and have found very useful ever since then: transformations of random variables. The basic idea is this: Given a random variable $X$ from which we know how it is distributed, i.e., either its probability density function (PDF) $f_X(x)$ or its cumulative distribution function (CDF) $F_X(x)$, find the distribution […]

More algebra fun

I recently posted on algebraic rules for rewriting linear and quadratic forms and showed how Kronecker products come into play when expanding products of matrices. Likewise, we might come into a situation where parameters of interest are inside a Kronecker product which we want to “pull out”. As the operation is still linear this should […]

Fun with algebra

Quite often, the key to finding an efficient solution to a given problem is to rewrite it in simpler terms. This requires a fair amount of experience and a good toolbox of rules one can apply to transform things. For me, problems are often algebraic and I have to deal with products of (tensors,) matrices […]