# Eigenthings (eigenvectors and eigenvalues) (Discussion)

Eigenthings (eigenvectors and eigenvalues) (Discussion)
0
#1

Eigenvectors and eigenvalues (alternative Simple English Wikipedia page) are a topic you hear a lot in linear algebra and data science machine learning.

However, these are very abstract terms and are difficult to understand why they are useful and what they really mean.

This forum post is to catalog helpful resources on uncovering the mysteries of these eigenthings and discuss common confusions around understanding them.

Here are some resources on the topic I have found useful:

Havenâ€™t looked through these, but look promising:

1 Like
#2

So, if I read the top answer from â€śWhat is the importance of eigenvalues/eigenvectors?â€ť correctlyâ€¦

a scalar is a scalar, unless all it does to a vector is stretch/compress/flip it, then it becomes an eigenvalue and the vector it stretches/compresses/flips becomes an eigenvector?

#3

Although I think I understand where youâ€™re coming from, I just want to be clear on when a scalar â€śbecomesâ€ť an eigenvalue.

I donâ€™t think you can just create eigenvalues, but rather they are a property of the matrix youâ€™re multiplying (which some describe as a linear transformation).

But to answer your question, it is a bit more nuanced the difference between a scalar and an eigenvalue. Yes, you are correct that an eigenvalue and scalar both stretch /compress/flip vectors. However, eigenvalues do these transformations (stretch/compress/flip) because they are actually (to my understanding) matrices that are being multiplied.

In exploring certain matrix multiplications, there are certain eigenvalues (with particular scalar amounts) with corresponding eigenvectors that have this special property of a simple stretch/compressing/flipping being the result of a matrix multiplication.

Thatâ€™s how I understand it and I hope that helps. If thereâ€™s anything inaccurate or confusing from my explanation, please point it out

#4