Normalization factor in multivariate Gaussian

Question

(This is possibly a silly question, but I am curious.)

The multivariate Gaussian PDF is typically written something like this

$$ \frac{1}{\sqrt{(2\pi)^{d}\lvert \boldsymbol\Sigma\rvert}} \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu}) \right) $$

where $d$ is the dimension of $\mathbf x$ (e.g. the above was taken from Wikipedia).

However it seems to me that the normalization factor could equivalently be written as $\sqrt{|2\pi\boldsymbol\Sigma|}$, letting the determinant take care of the implicit $d$ exponent. Moreover, this is simpler to write and gives a dimension-independent formula.

Is this an acceptable alternative notation?

Answer 1

Indeed the formula $$|2πΣ|=(2π)^d|Σ|$$ is correct.

In practice, one would compute $|Σ|$ and then multiply it by $(2π)^d$, rather than multiply $Σ$ by $2π$, which involves $d^2$ operations, and then compute its determinant.