nietdiagonalisierbare

nietdiagonalisierbare is a German term meaning “non-diagonalizable” and refers to matrices or linear operators that cannot be brought into a diagonal form via similarity transformation. In linear algebra, a matrix \(A\) is diagonalizable if there exists an invertible matrix \(P\) such that \(P^{-1}AP\) is diagonal. When no such transformation exists, \(A\) is non‑diagonalizable or, in German, nicht diagonalisierbar bzw. nichtdiagonalisierbare.

The most common cause of non‑diagonalizability is the presence of defective eigenvalues. If an eigenvalue’s algebraic multiplicity (the number of times it appears as a root of the characteristic polynomial) exceeds its geometric multiplicity (the dimension of its eigenspace), the matrix is defective and cannot be diagonalized. A simple example is the 2×2 matrix \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\), whose single eigenvalue 1 has algebraic multiplicity 2 but only one linearly independent eigenvector. The Jordan normal form of such a matrix contains at least one Jordan block of size greater than one.

Non‑diagonalizable operators still possess a rich structure. They can be expressed in Jordan canonical form, where each block accommodates generalized eigenvectors. These generalized eigenvectors satisfy \((A-\lambda I)^k v = 0\) for some minimal integer \(k\). In computational settings, non‑diagonalizable matrices can lead to numerical instability and require more sophisticated methods for functions of matrices, such as the matrix exponential.

The concept is essential in many areas of mathematics and physics, including differential equations, control theory, and quantum mechanics, where the diagonalizability of operators determines the simplicity of solutions and the behavior of dynamical systems. Non‑diagonalizable cases often signal resonances, defective modes, or symmetries that cannot be fully resolved by eigenvalue decomposition.