All simulators
Eigenvectors & Quadratic Forms
A symmetric matrix turns the unit circle into an ellipse whose axes are its eigenvectors.
Day 21: Eigenvalues, Eigenvectors & Quadratic Formsλ₁ (green)
3.000
λ₂ (violet)
1.000
v′Av
2.643
between λ₂ and λ₁
Type
positive definite
trace 4.00, det 3.00
λ₁ q₁λ₂ q₂probe vAv
The grey circle is every unit vector; the coloured ellipse is where A sends them. The eigenvectors are the ellipse's axes, and the eigenvalues are its semi-axis lengths. Rotate the probe v: Av lines up with v only along an eigenvector.