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I want to generate 3 random symmetric matrices, each of dimension 3 whose elements are normally distributed. It could be done simply as

A = RandomVariate[NormalDistribution[0, 1], {3, 3}]
B = RandomVariate[NormalDistribution[0, 1], {3, 3}]
C = RandomVariate[NormalDistribution[0, 1], {3, 3}]
R1 =(A + Transpose[A])/2
R2 =(B + Transpose[B])/2
R3 =(C + Transpose[C])/2

I am looking for other ways in which the same problem can be done for the case where number of matrices is large.

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  • $\begingroup$ If you have a recent version of Mathematica, there are a lot of built-in distributions for random matrices. See reference.wolfram.com/language/guide/MatrixDistributions.html for more. $\endgroup$
    – Pillsy
    Commented Apr 28, 2017 at 19:46
  • $\begingroup$ I can't see any problem with what you are doing - do you want a simple function that generates a random matrix? $\endgroup$
    – mikado
    Commented Apr 28, 2017 at 19:50
  • $\begingroup$ @mikado I want to generate 1000 random symmetric matrices for which using this code isn't a good idea. I am looking for better algorithm or a function to do so. $\endgroup$
    – NerdySnail
    Commented Apr 28, 2017 at 19:56
  • $\begingroup$ @NerdySnail Do you want just a few matrices (so you can hand pick the names) or a whole slew of them? $\endgroup$
    – Igor Rivin
    Commented Apr 29, 2017 at 15:43
  • $\begingroup$ Let's say we have 100 of them. Handpicking would be absurd. $\endgroup$
    – NerdySnail
    Commented Apr 29, 2017 at 16:26

2 Answers 2

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Firstly, your method does not work. The diagonal elements are $\mathcal{N}(0, 1),$ the off-diagonal elements are $\mathcal{N}(0, \frac{1}{\sqrt{2}}).$ Secondly, GaussianOrthogonalMatrixDistribution[\[Sigma],n] does work (in your case, $\sigma = 1, n=3.$

EDIT as pointed out by Mikey, the mathematica command does the wrong thing. However, this does:

randomSymMat = 
  Module[{mat = RandomVariate[#, {#2, #2}], upper, diag}, 
    upper = UpperTriangularize[mat, 1]; 
    diag = DiagonalMatrix[Diagonal@mat]; 
    diag + upper + Transpose[upper]] &;

so 

randomSymMat[NormalDistribution[0, 1], #]& /@ Array[3&, 1000]

Will do what you need.

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  • $\begingroup$ hmmm...reading the docs, GaussianOrthogonalMatrixDistribution[] is a distribution of a symmetric matrix (x+x^T)/2, where x is a square matrix with independent identically distributed matrix elements that follow NormalDistribution[0,[Sigma]]. Sure sounds like x is distributed same as A in his question, no? $\endgroup$
    – MikeY
    Commented Apr 28, 2017 at 20:19
  • $\begingroup$ @Igor,Indeed.As Pillsy already mentions,I should have used MatrixDistributions. However doing that too,doesn't lead us closer to the actual problem of generating a thousand of such matrices in a go. $\endgroup$
    – NerdySnail
    Commented Apr 28, 2017 at 20:21
  • $\begingroup$ @MikeY That is NOT the Gaussian Orthogonal Matrix Distribution, so either the name is wrong or the docs are wrong. $\endgroup$
    – Igor Rivin
    Commented Apr 28, 2017 at 20:22
  • $\begingroup$ @MikeY The docs are NOT wrong. This is really messed up. $\endgroup$
    – Igor Rivin
    Commented Apr 28, 2017 at 20:25
  • $\begingroup$ @IgorRivin You can report the problem at the Give Feedback link here: reference.wolfram.com/language/ref/… $\endgroup$
    – Alan
    Commented Apr 28, 2017 at 21:17
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OK, here's my shot at it.

makeMat[eps_, n_] := (res = RandomVariate[NormalDistribution[0, eps], {n, n}];
                     UpperTriangularize[res] + Transpose@UpperTriangularize[res, 1])

makeMat[1,3]

$$ \left( \begin{array}{ccc} 0.733676 & 0.0102509 & -0.35534 \\ 0.0102509 & -0.462317 & -0.132434 \\ -0.35534 & -0.132434 & 0.89037 \\ \end{array} \right) $$

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