Suppose I have a set of fixed vectors $\{f_n\}_{n=1}^N, \{g_n\}_{n=1}^N \subseteq \mathbb{R}^d$. If I want to build a matrix $M$ of size $d \times N$ with the following rule, let $M_i$ denote the $i$th column of $M$, then $$M_i = \begin{cases} f_i, \text{with probability } 1/2, \\ g_i, \text{with probability } 1/2. \end{cases}$$ Is there any literature on this type of construction? Could not find online, since the entries themselves are not random, but the way I choose my columns is.
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$\begingroup$ Of course this is equivalent to a particular way of selecting the entries randomly, just that the entries are not independent (which is usually the case for random matrices studied in random matrix theory). What do you want to know about this matrix $M$? $\endgroup$Qiaochu Yuan– Qiaochu Yuan2025-09-02 16:18:16 +00:00Commented Sep 2 at 16:18
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$\begingroup$ My question is can I use standard random matrix theory to study this matrix? $\endgroup$TheChosenOne– TheChosenOne2025-09-02 16:27:25 +00:00Commented Sep 2 at 16:27
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2$\begingroup$ As far as I know, standard random matrix theory nearly always considers the case that the entries of the matrix are independent. But, again, what do you want to know about this matrix $M$? It may not be too difficult to figure out directly. More specific questions are easier to work with. $\endgroup$Qiaochu Yuan– Qiaochu Yuan2025-09-02 16:47:32 +00:00Commented Sep 2 at 16:47
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