Eigenvalues and Eigenvectors
The special directions a transformation merely stretches — and the factors by which it stretches them — the hidden skeleton of a matrix.
Eigenvalues and Eigenvectors
For most vectors, a Linear Transformation both rotates and stretches. But almost every matrix has a few special directions it leaves pointing the same way, only scaled. A nonzero vector \vec v with
is an eigenvector of A, and the scalar \lambda is its eigenvalue — the factor by which that direction is stretched (|\lambda| > 1), shrunk (|\lambda| < 1), or flipped (\lambda < 0). Eigenvectors are the axes along which the transformation is purely a scaling, and finding them strips a complicated matrix down to its skeleton.
Finding them
The defining equation rearranges to (A - \lambda I)\vec v = \vec 0, which has a nonzero solution only when the matrix A - \lambda I collapses space — that is, when its Determinant vanishes:
This characteristic equation is a polynomial in \lambda; its roots are the eigenvalues, and back-substituting each one yields its eigenvector direction. For a symmetric matrix like A=\begin{bmatrix}2&1\\1&2\end{bmatrix}, the eigenvalues are 3 and 1, with perpendicular eigenvectors (1,1) and (1,-1).
Watch the dominant direction win
Apply a matrix over and over and a remarkable thing happens: almost any starting vector swings toward the eigenvector with the largest eigenvalue, because that direction grows fastest and drowns the others out. Below, a ring of vectors is repeatedly hit with A=\begin{bmatrix}2&1\\1&2\end{bmatrix} (and renormalized each step so they stay on screen). Watch them sweep onto the dominant eigenvector (1,1), the green line.
This iterate-and-normalize loop is the power method, and it is not a toy: it is precisely how PageRank finds the dominant eigenvector of a web's link Matrix to score every page at once.