Generate Random Symmetric positive definite matrix

Question

Is there any way to create a random symmetric positive definite matrix ?

Answer 1

See a-simple-algorithm-for-generating-positive-semidefinite-matrices

Translation to Mathematica:

n    = 5; (*size of matrix, change as needed*)
q    = Table[RandomReal[{-1, 1}], {n}, {n}];
mat  = Transpose[q].q;
PositiveDefiniteMatrixQ[mat]
(*true*)

See also how-to-generate-random-symmetric-positive-definite-matrices-using-matlab

The matrix symmetric positive definite matrix A can be written as , A = Q'DQ , where Q is a random matrix and D is a diagonal matrix with positive diagonal elements. The elements of Q and D can be randomly chosen to make a random A.

Here is the translation of the code to Mathematica

n         = 5; (*size of matrix. Change as needed *)
q         = Table[RandomReal[{-1, 1}], {n}, {n}];
eigenMean = 2; (*see link above *)

mat = Transpose[q].DiagonalMatrix[eigenMean + Table[RandomReal[{-1, 1}], {n}]].q;

PositiveDefiniteMatrixQ[mat]
(*true*)

Answer 2

dim = 10;

RandomVariate[GaussianOrthogonalMatrixDistribution[dim]]

or

With[{U = RandomVariate[CircularRealMatrixDistribution[dim]]},
 U.(RandomReal[{-1, 1}, dim] U\[Transpose])
 ]