Why do we care about Eigenvectors and Eigenvalues

Geometric Intuition

• Eigenvectors of a matrix are basically certain vectors that, when transformed by the matrix, can also be represented as a scalar multiple of the original vector.
• Since our new vector is a scalar multiple of the original vector, it must lie in the span.
• The Span of a single vector can be visualized as basically a line.
  • A single vector points a certain way.
  • The span of this one vector is literally the different ways we scale it.
• So our eigenvector, when transformed, lands back on the same line.
• The eigenvalue is the scalar multiple $\lambda$ such that $A\vec{x}=\lambda \vec{x}$. We’ll mostly deal with real valued $\lambda$s, an imaginary $\lambda$ signifies a rotation.

That is the key intuition behind eigenvectors: vectors that don’t get knocked off their Span after a transformation.

Why

• They seem to be useful in a lot of stuff.
• They make certain computations easier.
• Like for 3D rotations, the eigenvector is same as the axis of rotation–I guess this could be useful in robotics.