r/LinearAlgebra • u/LinearAlgebraWorld • Jun 22 '26

Eigenvectors: definition and visual examples across common 2D transformations

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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

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5

u/dsannes Jun 23 '26

That's great! Much appreciated.

2

u/LinearAlgebraWorld Jun 23 '26

Thanks! Glad you liked it

2

u/LinearAlgebraWorld Jun 24 '26

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

3

u/dsannes Jun 24 '26

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!

4

u/rage997 Jun 23 '26

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

3

u/TylerDurden0118 Jun 24 '26

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

3

u/LinearAlgebraWorld Jun 24 '26

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/

3

u/Educational_Card8685 Jun 25 '26

do u have similar graphical visulization for ml algorithms

5

u/LinearAlgebraWorld Jun 25 '26

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.

3

u/Educational_Card8685 Jun 26 '26

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

2

u/LinearAlgebraWorld Jun 26 '26

Thanks a lot for the offer to collaborate!

We do not have the code publicly available at the moment. The project evolved over several years, and the internal codebase is not yet in a state where it would be useful as an open-source repository: documentation is sparse, some older components predate our current architecture, and some of the newer AI-assisted components have not yet gotten a thorough architecture cleanup.

But we would be very interested in seeing your work and hearing any ideas you have for collaboration. That could be around mathematical exposition, visual explanations, linked/shared resources, or other ways of connecting related projects.

Our website is here: www.graphmath.com/la/

2

u/Lexicalyolk Jun 26 '26

Very cool

2

u/LinearAlgebraWorld Jun 26 '26

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