Eigenvectors
Directions a linear transformation stretches without turning away from their line.
Library note. Check the assumptions and further reading before applying a formula.
What Is This?
A matrix can rotate, stretch or mix the coordinates of a vector. An eigenvector is a nonzero vector that remains on the same line after the transformation. Its eigenvalue tells you the scale factor. A negative value reverses the direction. Eigenvectors let us study a complicated transformation through directions where its action is simpler.
Try an example
A matrix that doubles horizontal coordinates and triples vertical ones has the horizontal and vertical axes as eigenvector directions. Their eigenvalues are 2 and 3.
Where it needs care
A real matrix need not have real eigenvectors or enough independent eigenvectors to form a basis. A pure 90-degree rotation in the plane has no real eigenvector.
Historical Context
Eigenvalue problems grew from work on mechanics and differential equations. They became central to linear algebra and quantum mechanics.
Real-World Applications
- Find principal components of covariance matrices.
- Study vibration modes and stability of linear systems.
