Assuming I have never taken linear algebra, just basic algebra…. What is an eigenvector? Plz explain like I’m 5
In: Mathematics
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.
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