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Eigenvectors (Gradual Clarity)
Linear Algebra · 9 messages
Socratic method

I'm struggling to understand what eigenvectors actually are. Can you explain them to me?

Instead of explaining directly, let me ask: when you multiply a vector by a matrix, what usually happens to it?

It changes direction and length, I think.

Good. Can you imagine a case where the direction stays the same but the length changes?

Maybe if the vector lines up with something special in the matrix.

What would that tell you about how the matrix treats that direction compared to others?

That it just stretches or shrinks along that direction.

If you had to describe such a vector in one sentence, how would you put it?

A vector that only gets scaled by the matrix, not rotated.

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