Meaning of defective matrix | Babel Free

Definitions

A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable.

Examples

“1994, Sigal Ar, Jin-Yi Cai, Chapter 5: Reliable Benchmarks Using Numerical Stability, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, Society for Industrial and Applied Mathematics, page 41, An n x n matrix A is a defective matrix if it has a defective eigenvalue. Obviously, a Jordan block of dimension greater than 1, and a matrix whose Jordan canonical form has a Jordan block of dimension greater than 1, are defective matrices.”

“Therefore, the sum of the geometric multiplicities of a defective matrix will be less than n.”

“Thus, defective matrices exhibit high sensitivity to imperfections. It can be checked that the unperturbed matrices L* are in fact nearly-defective, because they have nearly-parallel eigenvectors. In particular, the matrices are themselves perturbations of order ε of exactly defective matrices L_(id).”

CEFR level