# What is the entropy of multivariate data multiplied by a vector?

It is a general rule that for multivariate data $$\boldsymbol{X}$$ and a matrix $$\boldsymbol{A}$$, their entropy is

$$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$ (see https://en.wikipedia.org/wiki/Differential_entropy#Properties_of_differential_entropy).

What is the rule for the entropy of $$\boldsymbol{X}$$ multiplied by a vector $$\boldsymbol{b}$$?

$$h(\boldsymbol{X} \boldsymbol{b} ) = ?$$

• Emulate the analysis given at stats.stackexchange.com/questions/415435. It's unclear what you mean by your "also" question, since all of this is explicitly about distributions.
– whuber
Sep 25, 2020 at 22:34
• thanks, edited the last part. the question here is what the non-reduced entropy solution is of multiplying a multivariate dataset by a vector of weights, each corresponding to a univariate component of the multivariate. Not about the effect of shift or scale. Your link is all about univariate entropy only Sep 25, 2020 at 22:55
• The rule you linked assumes X is a random vector, not a random matrix. So, by Xb, do you mean the inner product of X and b? Sep 26, 2020 at 1:15
• My link is about how to work with entropy. The method there extends, with no essential change, to multivariate problems.
– whuber
Sep 27, 2020 at 12:35