SimLabs

Eigenvectors & Eigenvalues

Directions that "stretch only, don't rotate" under transformation.

Move your mouse to find directions where the gray input vector aligns with the orange output vector.

Geometric motivation for diagonalization:
If we can find enough eigenvectors as a new basis, in this new coordinate system, matrix A becomes a "diagonal matrix" with values only on the diagonal. The transformation decomposes into simple axis-aligned stretches.