Problem 7

Draw 10000 samples from the set of positive definite symmetric \(5 \times 5\)-matrices with trace one. Use the distribution that Paul Breiding likes best. What is the expected value for the determinant of your matrices?

Solution

Mathematica

Make a random matrix with determinant 1:

RandomMatrix[n_] := Module[{upper, M},
  upper = RandomReal[{0, 1}, {n, n}];
  M = (upper + Transpose[upper])/(2 Tr[upper]);
  Return[M]
  ]

RandomMatrix[3] // MatrixForm

A module that tests if the matrix is positive semidefinite, add the determinants of those, and waits until we have done this M times:

ExpectedDet[M_] := Module[{totalmatrices, totalsofar, Mat},
  totalmatrices = 0;
  totalsofar = 0;
  While[totalsofar < M,
   Mat = RandomMatrix[5];
   If[
    PositiveDefiniteMatrixQ[Mat],
    totalsofar = totalsofar + 1;
    totalmatrices = totalmatrices + Det[Mat];
    ];
   ];
  Return[totalmatrices/totalsofar];
  ]

The result. The first number is how long it took in seconds, the second is the expected value of the determinant:

Timing[ExpectedDet[10000]]