diagonalization

Diagonalization rewrites a matrix in its simplest possible form. We factor A = P*D*P^(-1), where D is a diagonal matrix holding the eigenvalues and the columns of P are the matching eigenvectors. Read right to left, this says: switch into the eigenvector coordinate system, do nothing but independent stretches (that is D), then switch back.

Why bother? In the eigenbasis there is no tangled rotation left, only clean stretching along each axis. That makes hard jobs easy. Powers especially: A^k = P*D^k*P^(-1), and raising a diagonal matrix to a power just means raising each diagonal entry to that power. A messy repeated multiplication becomes a handful of scalar powers.

Honesty: not every matrix can be diagonalized. You need enough linearly independent eigenvectors to fill the columns of P. A matrix like [[1,1],[0,1]] has a repeated eigenvalue but only one direction of eigenvectors, so it falls short and cannot be diagonalized over the reals.