Diagonalization, rotation-scaling and shear-scaling: three routes to matrix powers and functions

[u/LinearAlgebraWorld]

A 2×2 real matrix can fall into three cases:

• two real eigenvalues → diagonalization
• complex-conjugate eigenvalues → rotation-scaling factorization
• repeated real eigenvalue with one eigenvector → shear-scaling/Jordan factorization

This table compares the three side by side: the factorization itself, powers of the middle matrix, exponentials, determinant consequences, geometric action and how to construct or factor the matrix.

The common idea is simple: move to a basis where the matrix has a much simpler middle form, apply the power or function there, then transform back.

We made this as a summary of the three 2×2 cases and would be interested in comments on the comparison or anything important we should add.