Soize uncertainty/randomness problem

[L.Richter]

Homework Statement

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 = LL^T

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.

Homework Equations

In general for 1D:

E[X] = μ_x
E[X²] = σ² + μ_x²
E[(X-μ_x)²] = σ_x²

The Attempt at a Solution

Compute G^-1:

G = LL^T

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

 

[KataKoniK]

Thank you for your question. it is important to approach any problem with a systematic and logical mindset. In this case, we are trying to compute the mean and second moment of G-1, a 2X2 matrix, in terms of the random elements of L.

To begin, we can start by defining L as a lower triangular matrix with elements L11, L21, and L22. Since L is lower triangular, we can assume that the off-diagonal elements are zero. Additionally, we are given that the diagonal elements are positive only.

Next, we can use the given information about L to determine the moments of G-1. From the homework equations provided, we know that the mean and second moment of a random variable X can be calculated using the mean and variance, respectively. Therefore, we can use the same approach to calculate the mean and second moment of G-1 in terms of the moments of the random elements of L.

For example, the mean of G-1 can be calculated as follows:

E[G-1] = E[(L-1)T(L-1)]

= E[(L-1)T]E[(L-1)]

= E[(L-1)]2

= E[L-1]2

= E[L-1]2 + E[L-1]2

= 2E[L-1]2

Similarly, the second moment of G-1 can be calculated as follows:

E[(G-1)2] = E[((L-1)T(L-1))2]

= E[((L-1)T)2]E[((L-1))2]

= E[(L-1)2]2

= E[L-1]2

= E[L-1]2 + E[L-1]2

= 2E[L-1]2

Using these equations, we can express the mean and second moment of G-1 in terms of the moments of the random elements of L. I hope this helps you in approaching the problem. If you have any further questions, please do not hesitate to ask. Good luck with your research!