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Reading this question, what is meant by “dimensionality” of x (denoted by letter $D$), where x is multivariate gaussian random variable?

Is it the same as rank? If so, why use different term? Is “dimensionality of a matrix” same thing as a rank?

Tomas
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    No it's the $n$ in the $n \times n$ matrix. You can see this because the trace of the identity is $D$. – CyclotomicField Apr 08 '25 at 15:12
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    $x$ is a vector, not a matrix. I'd interpret "dimensionality of $x$" in this case as the number of components in the vector, i.e. the dimension of the underlying vector space. Also, terminology is not universal - different authors sometimes say things in different ways. – Karl Apr 08 '25 at 15:18
  • @CyclotomicField thanks, however, could this not be the rank, in general? For example if $\Sigma$ wasn't full rank matrix (covariance matrix doesn't have to be, in general)... Then the tr() would sum up to the rank, no? Or perhaps this post works with full-rank matrices only (?) – Tomas Apr 08 '25 at 15:36
  • @Tomas an $n \times n$ matrix will not always have rank $n$. The dimensionality is the maximum possible rank the matrix could have, but the rank can be as low as $0$. – CyclotomicField Apr 08 '25 at 15:48
  • I second Karl's comment. Your title (and body) misled @CyclotomicField. – Anne Bauval Apr 08 '25 at 16:23
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    @Karl thanks, you are right, x is a vector! So the dimensionality of x is the length of vector x, in this particular context? – Tomas Apr 08 '25 at 17:54
  • @AnneBauval thanks, fixed the question. – Tomas Apr 08 '25 at 17:57
  • $X$ is a multidimensional random variable means its possible values are vectors with $n$ components, from a vector space of dimension $n$. If $X$ is Gaussian, then it can be generated as $a.U + x_0$, where $U$ is a normal, centered Gaussian variable, i. e. a random vector of dimension $D$ (with a diagonal, normalized variance tensor); $a$ is a constant linear operator ${\mathbb R}^D \longrightarrow {\mathbb R}^n$; $x_0$ is a constant, $n$-dim vector and the expectation of $X$. – François Jurain Apr 08 '25 at 19:30

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