Eigenvalue equation

Linear algebra · Formula · Appears in 1 lesson

Av=λv,v≠0A\mathbf{v} = \lambda \mathbf{v}, \quad \mathbf{v} \neq \mathbf{0}

Finds the directions a matrix leaves pointing the same way, stretching them by a single number. Those directions are the natural coordinate system of the map, which is what makes PCA and spectral methods work.

Symbols

AA
A square matrix.
v\mathbf{v}
An eigenvector — a direction the matrix preserves.
λ\lambda
The eigenvalue, the factor that direction is scaled by.
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