Thread by @akshay_pachaar

Eigenvalues & Eigenvectors clearly explained:

The concept of eigenvalues & eigenvectors is widely known yet poorly understood!

Today, I'll clearly explain their meaning & significance.

Let's go! 🚀

In linear algebra, eigenvalues and eigenvectors are ways of capturing the essence of linear transformations.

For any given transformation, we can represent it with a transformation matrix, denoted as 'A', which determines how vectors are transformed.

Here's how it works:

Now, imagine a transformation that changes the space but preserves some directions.

These special directions are the eigenvectors, and the scale of stretching or shrinking in these directions is given by eigenvalues.

Here's how we calculate them👇

Now, let's take a real example and put our statement to the test!

In the image below, pay close attention to how the transformation affects eigenvectors compared to regular vectors:

Physical Significance! 🌐

From Google's PageRank algorithm to PCA they're everywhere!

In PCA, eigenvalues quantify data variance captured by each principal component, while eigenvectors define the directions of maximum variance.

Implementation below uses eigenvector/values:

If you interested in:

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- LLMs/AI Engineering ⚙️

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