If you want to know the state after 100 steps, you'd need to calculate x 100 ​ =A 100 x 0 ​ . Computing A 100 by multiplying A by itself 99 times is very slow and computationally expensive.

max(Var u ​ )=max( u T C u )=λ 1 ​

The quantity you're interested in, u T C u , can be thought of as a score. This score measures how much of the stretched vector, C u , points back in the original direction of u .

The "randomness" in Random Forest comes from two key aspects: Bootstrap Aggregating (Bagging): Each individual decision tree in the forest is trained on a random sample of the training data, drawn with replacement. This means some data points may be used multiple times in the training of a single tree, while others may not be used at all. Feature Randomness: When splitting a node in a decision tree, the algorithm doesn't consider all available features. Instead, it randomly selects a subset of features and chooses the best split from that subset. This combination of bagging and feature randomness helps to create a forest of diverse and decorrelated trees, which reduces the overall variance of the model and makes it less prone to overfitting.