Understanding matrices intuitively, part 2, eigenvalues and eigenvectors - The Stata Blog

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Abstraction: Geometric intuition for eigenvalues and eigenvectors via matrix transforms

Key points:

Eigenvalues/eigenvectors solve Ax = λx: eigenvectors are the natural axes (lines through the origin) along which the transform Ax merely scales points rather than rotating them; eigenvalues record the scaling factor λ
A negative eigenvalue means the corresponding axis is flipped as well as scaled; eigenvalue of 1 means the axis is fixed under the transform
Near-singularity is indicated by a small ratio of smallest to largest eigenvalue (e.g., 0.07/2.78 = 0.03), not by small eigenvalues alone (a uniform compression has small eigenvalues but is not singular)
Computers tolerate extreme compression; practical singularity threshold is roughly 10^-8 of the original space area
Rotation matrices have no real eigenvectors because no line through the origin is preserved under rotation
A^-1 shares eigenvectors with A but has eigenvalues λ^-1; A^p has eigenvalues λ^p, enabling matrix square roots and fractional powers

Connections: Stata · Linear Algebra · Eigenvalues And Eigenvectors · Matrix Decomposition