What Is the Difference Between PCA and Karhunen Loeve (KL Transform)?
For discrete data both are the same - Finding set of orthogonal directions which maximizes the Variance (Energy) of data along them. Sometimes those are called the natural axis of the.
Since we're dealing with variance it is only natural both are calculated from the covariance matrix of data.

You may encounter places where KL might be even defined on the correlation matrix.

Anyhow, some nuances when dealing with them:


Usually when dealing with Continuous Function people use the term KL. PCA is preserved to discrete data (Though Mathematicians might work with the PCA in all cases as they have SVD for continuous data as well).
PCA is used for any discrete data where KL was specifically formed with random variables in mind.


In practice, when one has discrete data and tries to find the directions which maximizes the energy of the data (Dimensionality reduction, compression, etc...) both are the same.