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., .