Given are two random variables and . They can be represented as a vector . There holds . Using the KL expansion, find with .
Solution
Step 1: Find Mean & covariance
So the covariance matrix becomes:
Step 2: Find Eigenvalues and Eigenvectors
Since is already diagonal (uncorrelated random variables), the eigenvalues are simply the diagonal elements. :
The eigenvectors are:
- (corresponding to )
- (corresponding to )
important!: Ordered from largest to smallest So, and
Step 3: Construct the KL expansion
The KL expansion is given by:
where,
Therefore:
Simplifying:
Final KL expansion is:
Where and are independent standard normal random variables, i.e., .