An Intuitive Derivation of Eigenvectors

If you want to get an intuitive, visual understanding of linear algebra -- including eigenvectors&#x2F;eigenvalues -- 3blue1brown&#x27;s playlist on the subject is just ... perfect. <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=fNk_zzaMoSs&amp;list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=fNk_zzaMoSs&amp;list=PLZHQObOWTQ...</a>

really good explanation. I like it even better than 3blue1brown or visualisations that I had seen.<p>It is better because it really covers every step of the construction process.<p>And offers explanation of why certain thing are not the right construction blocks. The author gives a visual example, for example, of why basic vectors 1,0 -1,0 are bad. The article shows they cannot span the whole space.<p>Those kinds of explanations of &#x27;bad constructions&#x27; are difficult to show in visualizations, that show &#x27;good&#x27; constructions only.<p>But, yet, in my view, these negative examples, are really helpful to explain the material that otherwise, requires &#x27;intuition&#x27;.<p>Not everybody has same intuition, so showing negative examples&#x2F;impossible constructions, and why those do not work -- is a good way tuning one&#x27;s intuition.<p>---<p>On a separate note, I am wondering if such good step by step + counter examples, knowledge presentation -- is a result of author studying at MIT, or a natural trait (or both) ?

One way to think of eigenvectors:<p>Your linear map A moves things around, and you aim to characterise the linear map.<p>So, look for lines (through the origin) that are <i>not</i> moved. Those are given by eigenvectors. A point on that line might be moved closer to or further away from the origin (depends on eigenvalue &lt; or &gt; 1), or even flipped to the other side (if eigenvalue &lt; 0), but the line as a whole is mapped to itself.

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I also really liked this explanation: <a href="http:&#x2F;&#x2F;setosa.io&#x2F;ev&#x2F;eigenvectors-and-eigenvalues&#x2F;" rel="nofollow">http:&#x2F;&#x2F;setosa.io&#x2F;ev&#x2F;eigenvectors-and-eigenvalues&#x2F;</a><p>You can drag things around and change values -- if you&#x27;re a visual learner, it really helps grasp things like this.

IMO eigenvectors are easiest to understand in connection with differential equations, and that&#x27;s also one of their most important applications. If you plot the flow of the equation x&#x27; = Ax then the eigenvectors are visually apparent.<p><a href="https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input&#x2F;?i=stream+plot+%7B-5x+%2B+3y,+3x+%2B+8y%7D" rel="nofollow">https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input&#x2F;?i=stream+plot+%7B-5x+%2B...</a><p>The eigendirections are the directions where the solution moves in a straight line.<p>Not all matrices have (real valued) eigenvectors:<p><a href="https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input&#x2F;?i=stream+plot+%7Bx+%2B+5y,+-x+%2B+y%7D" rel="nofollow">https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input&#x2F;?i=stream+plot+%7Bx+%2B+5...</a>

The difference between an exposition in text(like this one) and videos(3blue1brown) is how people <i>prefer</i> to build knowledge and more deeply with how one learns. The quality of both of these is excellent. And one can test what works best like explaining to oneself(or a rubber duck) after reading&#x2F;viewing the material and how one can recall things.

There is no shortage of intuitive explanations of various elementary concepts in mathematics (and physics); my personal favorites are by E. Khutoyansky [1]. (Surely enough, there is a video on eigenvectors!)<p>[1] <a href="https:&#x2F;&#x2F;m.youtube.com&#x2F;user&#x2F;EugeneKhutoryansky" rel="nofollow">https:&#x2F;&#x2F;m.youtube.com&#x2F;user&#x2F;EugeneKhutoryansky</a>

I liked the MIT lecture on eigenvectors: <a href="https:&#x2F;&#x2F;ocw.mit.edu&#x2F;courses&#x2F;mathematics&#x2F;18-06sc-linear-algebra-fall-2011&#x2F;least-squares-determinants-and-eigenvalues&#x2F;eigenvalues-and-eigenvectors&#x2F;" rel="nofollow">https:&#x2F;&#x2F;ocw.mit.edu&#x2F;courses&#x2F;mathematics&#x2F;18-06sc-linear-algeb...</a>

The thing I use to visualize an eigenvector is exactly that. A rotating planet. The eigenvector of the rotation matrix being the axis of rotation.<p>It gets weird when thinking of 2D rotations though... Too complex for me!

Good writeup! A tad bit disappointed as I was expecting some new insights, but this is more or less standard material about eigenvectors you would get in a typical university course.

I really think some better name than eigenvectors (and eigenvalues, etc..) should be popularized. I find them to be very obtuse and opaque terms.

this is awesome tyvm!