Eigenvectors

An eigenvector stays on its own line when a matrix acts. It only stretches or shrinks, and the stretch factor is λ.

The red arrows lie on two lines. One line stretches by 3. The other stretches by 1. The ink arrows turn off their lines.

a 2.00 b 1.00 c 1.00 d 2.00

λ13.00
λ21.00
Linestwo

The matrix is the block of sliders, row by row: a b, then c d. A negative λ flips the arrow along that same line. A turn has no real eigenvector, so every arrow leaves its line.

2×2 matrix. Read each row from left to right. The columns move the arrows.
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Where this is used

The original PageRank method ranks a web page with an eigenvector of the link matrix. A search engineer reads that score as the page's importance.
In data science, principal component analysis finds the directions where measurements spread the most. Those directions are eigenvectors. A data scientist uses them to simplify a table.
A structural engineer finds the vibration modes of a building or a bridge. Each mode is an eigenvector. The matching factor sets how fast that mode shakes.
A mechanical engineer finds the principal axes of stress. They are the eigenvectors of the stress matrix, in three dimensions. Pull and push act along those axes.

Common mistakes

Most arrows are not eigenvectors. The ink arrows turn off their lines.
A pure turn has no real eigenvector. Every arrow leaves its line.
λ is the stretch along that line. It is not one of the four matrix entries by itself.
A negative λ flips the arrow along the same line. It does not send the arrow onto a new direction.

Inverse · Lessons