Assuming I have never taken linear algebra, just basic algebra…. What is an eigenvector? Plz explain like I’m 5
In: Mathematics
Imagine a really complicated series of train tracks that go every which way crossing a fault line where there are a lot of earthquakes. Just tons of tracks going all over the place.
One day there is an Earthquake and the ground shakes and all the train tracks move around.
Let’s agree every train track experiences the same Earthquake motion.
Almost all the train tracks get pulled and stretched and broken as the ground shifts.
One special track, though, was in just the right spot where the Earthquake *pulled it* along the direction it was pointed in anyway, so it just got a teeny bit longer. In a sense, its the only track that survived the Earthquake without changing shape, being bent, or breaking, but it’s length did change slightly.
What was special about that train track was both, the exact type of Earthquake that happened, and the direction the track happened to be in in the first place, those two qualities “paired up” to make a train track that didn’t bend or break, it just got **longer**. *That’s an Eigenvector*.
In math we often do things called “Linear Transformations” that are like my Earthquake, they take old lines and shift them into new lines. Some old lines though, are perfectly paired up to that specific transformation, that specific earthquake, and they won’t shift into a new direction afterwards, they’ll just be stretched or squashed down by it, and we call those the “Eigenvector” of that transformation.
Imagine straight evenly spaced grid lines filling a 2D space.
Now imagine that all the points in that space are moved according to a rule. The one catch is that the grid lines must stay straight and evenly spaced. This is called a linear transformation.
Now imagine a straight line. When you play a linear transformation, the straight line may rotate or shift, but it will still be a straight line.
Now imagine a line that doesn’t rotate or shift at all. Perhaps the points inside the line move to another point on the line, but the line itself is maintained. The points on this line are called “eigen vectors”.
Eigen Vectors are interesting, because when you apply a linear transformation, the eigen vectors don’t go through a complicated change. The eigen vectors are just scaled by a special number (called an eigen value)
Linear transformations are easiest to manage when we focus on the eigen vectors (where the behaviour is simple). Figure out all the eigen vectors and all the eigen values and you have a complete understanding of the linear transformation without needing to look at any of the complicated bits.
I hope this analogy/post is allowed… don’t tell this to a literal five year old.
When my manager/producer asked me this question, I said take a condom and a sharpie. Magnum size is easier to work with. Wash it. Draw one arrow from the base to the tip. Draw another arrow in a circle that goes around the shaft. Draw a third arrow that from the base that is sort of diagonal between the other two arrows. Now stretch the condom length wise, the first arrow will get longer but will not change the direction. Now stretch it horizontally, the second arrow will stretch but not change the direction. The third arrow will change the direction in both cases. The arrows that don’t change the direction are your eigenvectors (under a particular linear transform), the amount of increase in length are your eigenvalues. These come in pairs eigenvalue/eigenvector. the larger the eigenvalue the more important the eigenvector is. I guess it is more important to have a long (censored) than a thick one under this logic.
If I tell you to take a toy cube and say “you can turn this around and stretch it however you want” then you will have some axes about which you can turn it and some directions in which you will strech it. The arrows pointing in the direction of streching or the axes about which you turn are the eigenvectors. If you turn it about multiple axes, and strech it in multiple directions, there are some simple rules about how you have to add those up.
a vector that minds its own business even when everyone else is digging their nose into others’.
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