# Soize uncertainty/randomness problem

1. Apr 11, 2012

### L.Richter

1. The problem statement, all variables and given/known data

Following a maximization of the entropy, Soize obtained that random mass, stiffness, and damping matrices would involve a positive definite random matrix G, expressed in Cholesky form

G = LLT

where L is a lower triangular matrix. All elements of L are independent of each other, which allows the determination of the moments of G.

It is desired to compute the mean and second moment of G-1 for a 2X2 matrix G in terms of appropriate moments of the random elements of L. Note: off-diagonal elements of L are zero mean random variables (Gaussian) and the diagonal elements of L are positive only.

2. Relevant equations

In general for 1D:

E[X] = μx
E[X2] = σ2 + μx2
E[(X-μx)2] = σx2

3. The attempt at a solution
Compute G-1:

G = LLT

G-1 = (LT)-1(L)-1

G-1 = (L-1)T(L-1)

My assumption is that I'm trying to find the moments, mean and second, for L11, L21 and L22, the elements of the lower triangle of G-1 which is a 2X2 matrix.

I have no definition for L at this time. I'm still looking in research papers. How should I approach the problem? Thanks in advance for any suggestions.

Last edited: Apr 11, 2012