英和和英

名詞

defective matrix (複数形 defective matrices)

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

   • 2007, Thomas S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, page 261,

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

   • 2013, Angelo Luongo, Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures, Alexander F. Vakakis (editor), Normal Modes and Localization in Nonlinear Systems, Springer, page 153,

     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.