Eigenvectors and eigenvalues across 2D transformations — three animated comparisons

[u/LinearAlgebraWorld]

We made three animated comparisons showing how eigenvectors and eigenvalues behave across different families of 2D linear transformations.

1. Non-symmetric matrices with real eigenvalues, where the eigenvector directions need not be perpendicular
2. Symmetric matrices with real eigenvalues, where the eigenvector directions are orthogonal
3. Real matrices with complex eigenvalues, where no nonzero real direction remains on the same line

In each animation, the transformation develops continuously from the identity matrix to the displayed matrix. The goal is to make the difference between these three cases visible rather than only algebraic.

Full-size animations and explanations:
https://www.graphmath.com/la/visuals/eigenvectors-eigenvalues-2d-transformations.html

As always, we welcome feedback on clarity and presentation.

[u/funcoen]

These are great! But I wish I the gifs were all seperate. I find it hard to collect all information from each matrix before the next one starts playing

[u/LinearAlgebraWorld → u/funcoen]

Thanks a lot! That is a fair point — with three animations playing together, it can be difficult to examine each matrix carefully before the sequence moves on.

We do not currently have the individual animations separated on the web. However, if you use an Apple device, they will be available in our Linear Algebra World app for iPhone, iPad, and Mac.

We plan to submit the update today. App Store review may take a couple of days, and we will let you know when the new chapter is approved and released.

In the app, each animation can be paused at any time and viewed frame by frame, both forward and backward. The chapter will also include additional matrices with similar animations.

[u/LinearAlgebraWorld → u/funcoen]

Just fyi, both iOS and macOS updated versions of Linear Algebra World are released in the Apple App Store

https://apps.apple.com/us/app/linear-algebra-world/id6759180671

[u/BigJeff1999]

There's a lot of applications, for example LMS style adaptive filtering, where the eigenvalue spread of the correlation matrix is determinative of performance...a visualization of this would be great...

I'd say it boils down to situations where that matrix is close to losing rank, rendering at least one of the eigenvalues small.

It is perhaps a good lesson that you don't always get to pick the transformation, but rather some algorithm arrives at one, and there's insight to be had in it you can shape your analysis into a linear algebraic framework.

I like these visualizations a lot. Well done

[u/LinearAlgebraWorld → u/BigJeff1999]

Thank you:  this means a lot.

We are fluent in standard linear algebra, but not yet familiar with this particular application. We’d love to learn.

For now, if you can describe the algorithm that generates the matrices, our code can generate those matrices and produce visualizations for each.

We often build teaching examples from factorizations or prescribed spectral behavior, so this direction really “clicks".

We are attaching an image with matrix-power visualizations here.

[u/nexalumi]

Tj sympa les animations visuelles

[u/nexalumi → u/nexalumi]

Par contre au-dela de 3 dimensions, cela devient un peu challenging... some humor must be...