If $X$ is a matrix valued random variables, are its singular values independent as random variables?
2026-03-26 12:51:58.1774529518
Independence of singular values
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Generally not, but it depends on the distribution of $X$. Also you should clarify what you mean. For example, if we take (as is often done) the singular values sorted in decreasing order, this ordering would usually make them dependent: if $A$ and $B$ are independent random variables such that $A \ge B$ a.s., then there must be a constant $c$ such that $A \ge c \ge B$ a.s. On the other hand, if $X$ is diagonal with independent diagonal elements, the singular values $|X_{ii}|$ listed in order of $i$ are independent.