eigenvalue (spectral) factorization

The eigenvalue factorization is diagonalization seen as a factorization. If A has a full set of linearly independent eigenvectors, gather them as the columns of X and put the corresponding eigenvalues on the diagonal of Lambda; then A = X Lambda X^-1. Reading right to left: change into the eigenvector coordinates, scale each axis by its eigenvalue, change back.

Once you have this, powers and functions of A become trivial. A^k = X Lambda^k X^-1, since the inner X^-1 X factors cancel, so raising A to a power is just raising each eigenvalue to that power. The same idea defines the matrix exponential e^A = X e^Lambda X^-1, which solves linear differential equations. The factorization turns hard matrix questions into easy scalar questions on the diagonal.

The decisive caveat is that X^-1 must exist, that is, A must have n independent eigenvectors. Defective matrices (those short on eigenvectors, like a nontrivial Jordan block) have no such factorization at all. Even when X exists, it can be nearly singular, with a huge condition number, making the factorization numerically treacherous to use.

Contrast this with the orthogonal spectral form. When A is symmetric (or Hermitian), the spectral theorem upgrades X to an orthogonal/unitary Q, giving A = Q Lambda Q^*, with real eigenvalues and perfectly conditioned, orthonormal eigenvectors. The general eigenvalue factorization is the fragile cousin; the symmetric spectral decomposition is the robust ideal.