Random symmetric positive-definite matrices

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This section describes the generation of random symmetric positive-definite (SPD) matrices.

With a given determinant

The problem is to generate a random symmetric positive definite matrix `S in RR^{n xx n}` with `det(S) = d`. Let `Q in RR^{n xx n}` be a uniform random orthogonal matrix, `D in RR^{n xx n}` be a positive diagonal matrix, and

Then `S` is symmetric:

`S` is positive definite:

and

Conversely, all symmetric positive definite matrices `S` can be brought to the form of `S = QDQ^T` by eigen-decomposition. The generation of a random orthogonal matrix `Q` is described in Uniform random orthogonal matrix. Now we just need to generate a diagonal matrix `D` such that `det(D) = d`.

Random diagonal matrix with given positive determinant

Let

and let

be a set of n uniform random numbers in `[0, -ln(d)]`. Sort this list to ascending order with a permutation `p`. Define

Then

We claim `D` can be formed by setting

To check this claim, compute:

The diagonal elements of `D` are also positive.

With given determinant and condition number

The problem is to generate a random symmetric positive-definite matrix `S in RR^{n xx n}` with `det(S) = d` and `cond(S) = c`. Let `Q in RR^{n xx n}` be a uniform random orthogonal matrix, `D in RR^{n xx n}` be a positive diagonal matrix, and

This matrix has the properties as above. Now we just need to generate `D` such that `det(S) = d`, and `cond(S) = c`. Let

and assume `a >= b`. Then

and

Thus we get the pair of equations:

Now

and

Therefore

and

Thus the solution is:

We assumed

This is the only restriction.

Remark 1.7.5 - Page generated 16.06.2017 01:36.