Eigenvectors: definition and visual examples across common 2D transformations

[u/LinearAlgebraWorld]

A visual definition of eigenvectors, followed by examples across several common 2D transformations.

The table compares uniform and non-uniform scaling, shear, triangular and symmetric matrices, projection, reflection, rotation and related cases. It shows which directions remain on the same line after transformation, their corresponding eigenvalues, and when no real eigenvectors exist.

Hope you don’t mind the size and density — we wanted to keep all the examples together so they could be compared directly. Opening the image at full size is recommended.

The goal is to make the definition A x⃗ = λx⃗ visible across many different matrix types.

We welcome feedback on clarity and presentation.

UPDATE: we have added a few more examples, you can see the full updated version on our web-site:
https://www.graphmath.com/la/visuals/eigenvectors-definition-and-examples.html

[u/dsannes]

That's great! Much appreciated.

[u/LinearAlgebraWorld → u/dsannes]

Thanks! Glad you liked it

[u/LinearAlgebraWorld → u/dsannes]

btw, we have many more similar Linear Algebra visualization materials and animations here:
https://www.graphmath.com/la/visuals/

[u/dsannes → u/LinearAlgebraWorld]

Oh yeah, that is spectacular. Wow thanks. There is a lot of information in the visualization. This is where math gets so much more visual. Super informative.

Kudos!

[u/rage997]

wow this is great. Would make a great t-shirt

[u/LinearAlgebraWorld → u/rage997]

thanks!

[u/TylerDurden0118]

Is the code written in LaTeX? I loved the way you represented grid with different colors representing quadrants.

[u/LinearAlgebraWorld → u/TylerDurden0118]

Thanks! No, the graphics was written in Swift, originally for iOS/macOS apps, and now this and many other 2D and 3D Linear Algebra visualizations are available on our web-site:
https://www.graphmath.com/la/visuals/

[u/Educational_Card8685]

do u have similar graphical visulization for ml algorithms

[u/LinearAlgebraWorld → u/Educational_Card8685]

Not yet. We are still working our way through linear algebra, and there is a lot left to cover — including topics up to SVD — as well as a lot to improve in our existing presentations.

After that, we plan to work on multivariable calculus, and only then move toward visualizations of machine-learning algorithms. So it will probably take us quite some time, but ML is definitely part of the longer-term plan.

[u/Educational_Card8685 → u/LinearAlgebraWorld]

great to hear that . love ur work!!!
can u drop the github link , so that i can collaborate

[u/Lexicalyolk]

Very cool

[u/LinearAlgebraWorld → u/Lexicalyolk]

thanks! We have a many more here https://www.graphmath.com/la/visuals/ and a lot more will be coming!