Linear Algebra

Eigenvectors & Eigenvalues

A matrix knocks almost every vector off its own line. Aim v anywhere and see whether Av stays on the dashed line through v: where it does, v is an eigenvector, and how far it stretches is the eigenvalue.

v and the line through itAv, where A sends iteigen-lines: A keeps them in place

Drag anywhere to aim v

How far Av is off v's line—

—

Matrix A

1.500.800.401.10

a 1.50

b 0.80

c 0.40

d 1.10

Presets

Direction of v 60°

Sweep I → A 1.00

View

Show

Where A sends every directionThe unit circle and its image

Values

eigenvalues λ—

eigen-directions—

turn of v—

|Av| / |v|—

trace, det—

What to observe

Aim v around the circle. Av almost always lands off the dashed line through v. Where it lands on it, nothing turned: Av = λv, and v is an eigenvector.
λ is how much v is stretched along its line. On Flip one λ is negative: Av stays on the line but points the other way.
Shear keeps only one line, and Rotate keeps none: every direction gets turned, so there is no real eigenvector.
Press Replace v by Av a few times. Whatever v you start from, it swings onto the eigen-line with the biggest |λ|. That is howPCA finds its first component.

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